IN  MEMORIAM 
FLOR1AN  CAJORI 


v-<y 


N  OT  ES 


ELECTRICITY  AND  MAGNETISM. 


DESIGNED  AS  A  COMPANION  TO  SILVANUS  P.  THOMPSON'S 
ELEMENTARY   LESSONS. 


BY 

J.    B.    MURDCCK, 

LIEUTENANT   U.    S.    NAVY. 


MACMILLAN    &    CO, 

1891. 


COPYRIGHT,  1883, 
BY  MACMILLAN  &  CO. 


/1$ 


PREFACE. 


THE  design  of  this  small  volume  of  notes  is  to  supple- 
ment the  instruction  given  in  Prof.  Thompson's  admirable 
Elementary  Lessons  by  such  explanations  and  additional 
instruction  as  an  experience  with  two  classes  of  cadets  in 
this  institution  has  shown  to  be  necessary.  In  general, 
notes  have  been  made  on  separate  paragraphs  in  the  les- 
sons, but  it  was  thought  best  to  treat  many  subjects  inde- 
pendently in  order  to  present  them  more  connectedly,  and 
it  is  hoped  that  the  notes  may  thus  be  of  some  service  by 
themselves.  The  endeavor  has  been  made  to  trace  the 
theory  of  the  dynamo  machine  and  of  electric  motors  from 
the  primary  laws  of  electro-magnetic  induction,  and  de- 
scriptions of  several  of  the  more  important  dynamo  ma- 
chines have  been  added,  chiefly  to  illustrate  the  various 
applications  of  the  general  principle  underlying  all.  As 
Prof.  Thompson's  treatise  has  come  to  be  largely  used  in 
colleges  and  high  schools,  demonstrations  by  the  aid  of 
calculus,  most  of  which  have  been  used  in  the  course  in 
this  institution,  have  been  given  to  replace  the  geometrical 

proofs  of  the  lessons  if  desired. 

J.  B.  MURDOCK. 
U.  S.  NAVAL  ACADEMY, 
Annapolis,    Md.,  July  10,  1883. 

NOTE.— Reference  have  been  made  in  the  text  to  the  figures  in  the 
Elementary  Lessons,  as  well  as  to  those  in  this  volume.  The  latter  range 
from  i  to  38,  and  higher  numbers  are  to  be  understood  as  referring  to 
those  in  the  Elementary  Lessons. 


M304876 


INDEX   TO    NOTES. 


I.  GALVANOMETERS. 

NOTE  PAGE 

1.  Tangent  galvanometer I 

2.  Sine  galvanometer 3 

3.  Mirror  galvanometer 4 

4.  Differential  galvanometer 4 

5.  Ballistic  galvanometer 6 

II.  THEORY  OF  POTENTIAL. 

6.  Illustration  of  meaning  of  potential 8 

7.  Difference  between  work  and  potential 10 

8.  Positive  and  negative  work ri 

9.  Positive  electricity  always  flows  from  a  high  to  a  low 

potential _i  I 

10.  Units  of  potential  and  work 12 

11.  Formula  for  electrostatic  potential 13 

12.  Zero  potential 14 

13.  Difference  of  potential 14 

14.  Electric  force 15 

15.  Law  of  inverse  squares 15 

16.  Capacity . , 16 

17.  Unit  of  capacity 17 

18.  Electric  force  exerted  by  a  charged  plate 17 

19.  Dimensions  of  units    18 

20.  Attracted  disc  electrometer 20 

21.  Absolute  electrometer. .  20 


vi  INDEX    TO   NOTES. 

III.  THEORY  OF  MAGNETIC  POTENTIAL. 

NOTE  PAGE 

22.  Magnetic  field 23 

23.  Method  of  mapping  a  field  by  lines  of  force 24 

24.  Equipotential  surfaces 26 

25.  Lines  of  force  due  to  a  single  pole 26 

26.  Lines  of  force  due  to  a  current 28 

27.  Magnetic  potential 29 

28.  Tubes  of  force.    ...    30 

29.  Intensity  of  magnetization 31 

30.  Solenoidal  magnets 32 

31.  Potential  due  to  a  magnetic  shell 33 

32.  Equipotential  surfaces  and  lines  of  force  of  a  magnetic 

shell 35 

33.  Work  done  in  moving  a  pole  near  a  shell 36 

34.  Equivalent  magnetic  shells 37 

35.  Potential  due  to  a  closed  voltaic  circuit 39 

36.  Work  done  in  moving  a  circuit  near  a  pole 40 

37.  Intensity  of  field  due  to  a  voltaic  circuit 42 

38.  Position  of  equilibrium  of  a  circuit  and  magnet 43 

39.  Mutual  potential  of  two  circuits 44 

40.  Conversion  of  units.      Tables  of  absolute  and  practical 

units 45 

41.  Determination    of   the  horizontal  component   of   the 

earth's  magnetism 48 

IV.  MEASUREMENTS  AND  FORMULAS. 

42.  Solenoids.    Ampere's  theory  of  magnetism 54 

43.  Best  arrangement  of  cells 5° 

44.  Long  and  short  coil  galvanometers 57 

45.  Divided  circuits 5& 

46.  Shunts 59 

47.  KirchhofFs  laws 61 

48.  Fall  of  potential 62 

49.  Wheatstone's  bridge 64 

50.  Proof  of  theory  of  bridge  by  Kirchhoff's  laws 65 


INDEX  TO  NOTES  vii 

NOTE  PAGE 

51.  Measurement  of  electromotive  force 66 

52.  Measurement  of  internal  resistance 68 

53.  Measurement  of  the  capacity  of  a  condenser 70 

54.  Determination  of  the  ohm 71 

55.  Practical  electromagnetic  units  of  heat  and  of  work. . .  72 

V.  ELECTRIC  LIGHTS. 

56.  The  voltaic  arc 75 

57.  Arc  lamps 77 

58.  Incandescent  lamps 79 

VI.  ELECTRO-MAGNETIC  INDUCTION. 

59.  Induction  currents  produced  by  currents 82 

60.  Determination  of  the  induced  electromotive  force 84 

61.  Practical  rule  for  the  direction  of  induced  currents. ...  86 

62.  Lenz's  law 87 

63.  Mutual  induction  of  two  circuits 88 

64.  Self  induction 88 

65.  Helmholtz's  equations 89 

66.  Induction  coil 90 

VII.  DYNAMO  MACHINES. 

67.  General  principles. ...  93 

68.  Electromotive  force 97 

69.  Efficiency IO2 

70.  Electromotive  force  in  circuit 105 

71.  Siemens'  machine 106 

72.  Gramme  machine 107 

73.  Brush  machine 109 

74.  Edison  machine 114 

75.  Alternate  current  machines 1 15 

VIII.  ELECTRIC  MOTORS. 

76.  General  principles II? 

77.  Electric  transmission  of  power  to  a  distance 118 


vin  INDEX  TO   NOTES. 


NOTE  PAGE 

78.  Theory  of  electric  motors ng 

79.  Modifications  of  theory  in  practice 125 

80.  Peltier  effect 127 

81.  Secondary  batteries 128 

IX.  TELEGRAPHY  AND  TELEPHONY. 

82.  The  Morse  alphabet 130 

83.  American  system  of  telegraphy 131 

84.  Faults. 134 

85.  Simultaneous  transmission 134 

86.  Blake's  transmitter 137 

87.  Telephone  exchanges 138 


NOTES 

ON 

ELECTRICITY  AND  MAGNETISM. 


I.  GALVANOMETERS. 

(Thompson's  Electricity,  pages  163-171.) 

i.  Tangent  Galvanometer  (§  199). 


To  derive  the  formula  in  g  200. 
fleeted  by  the  current  to  an 
angle  0  with  the  meridian  and 
be  in  equilibrium.  The  sum 
of  the  moments  of  the  forces 
acting  on  it,  taken  around  its 
axis,  is  then  zero.  The  moment 
tending  to  bring  it  back  into 
the  meridian  is  the  force  mH 
into  the  arm  BC,  or  mlHsm  B, 
m  being  the  strength  of  mag- 
net pole  and  /  the  distance  be- 
tween the  poles.  The  moment 
of  the  deflecting  couple  is  fm 
x  DE,  or  fml  cos  0,  f  being 
the  deflecting  force  of  the  cur- 
rent. If  now  the  magnet  be 
so  small  that  in  all  positions  its 
poles  may  be  considered  to  be 
at  the  centre  of  the  coil/  = 


Let  the  magnet  be  de- 
H 


^     @  195).    If  the  coil  has  n  turns,  each  exerting  this 


NOTES   ON 


force  on  a  pole  at  the  centre,  f  =  -     — .    Substituting 

this  value  of  /  and  equating  the  moments  of  the  forces 
around  A 

•27tnC 
mtffsm  0=  ml  cos  Q 


.-.  C  =  —  /f  tan  0. 


This  formula  shows  not  only  that  currents  vary  as  the 
tangent  of  the  angle  of  deflection,  but  gives  the  value  of 
the  currents  in  the  absolute  units  of  current  defined  in 
g  196.  The  practical  unit  of  current,  the  ampere,  being 
only  iV  of  the  absolute  unit,  the  result  must  be  multi- 
plied by  ten  to  obtain  the  current  in  amperes  from  any 

observed  deflection.     The  fraction  -  depending  on  the 

construction  of  the  instrument,  is  called  the  "  reduction 
factor  "  of  the  galvanometer.  It  is  generally  furnished  by 
the  maker,  but  may  be  readily  determined  by  electrolysis. 
If  an  electrolytic  cell  and  a  galvanometer  are  in  the  same 
circuit,  as  the  current  is  the  same  throughout,  the  value 
of  C  in  amperes,  as  given  by  the  tangent  galvanometer, 
may  be  equated  with  that  from  the  equation  on  page  177. 
Thus 


L.  //tan  6 
zt        zitn 


.'.  Reduction  factor  =  —  -  -. 
Hzt  tan  6 

The  demonstration  given  above  shows  that  the  deflec- 
tion is  independent  of  the  strength  of  pole,  and  it  is  not, 
therefore,  necessary  to  magnetize  the  needle  strongly.  As 
the  demonstration  involves  a  value  of/  which  is  true  only 
when  the  pole  is  at  the  centre  of  the  coil,  the  magnet 


ELECTRICITY  AND   MAGNETISM.  3 

must  be  so  small,  relatively  to  the  radius,  that  neither  of 
its  poles  should  in  any  position  depart  widely  from  the 
centre.  As  the  lines  of  force  due  to  a  current  in  a  coil 
pass  through  its  plane  perpendicularly,  as  shown  in  Fig. 
86,  the  dimensions  of  the  needle  and  the  accuracy  of  the 
instrument  may  be  increased  by  using  two  coils  with  the 
needle  midway  on  their  common  axis.  The  lines  of  force 
of  the  two  coils  then  act  together  so  that  they  are  sensi- 
bly parallel  in  the  region  in  which  the  needle  moves, 
many  of  them  passing  through  both  coils  perpendicularly. 

2.  Sine  Galvanometer  (§  201). 

It  is  possible  to  construct  a  sine  galvanometer  to  meas- 
ure currents  in  absolute  units,  but  the  ordinary  form  of 
the  instrument  is  not  intended  for  absolute  but  for  rela- 
tive values  only.  The  coil  is  placed  parallel  to  the  needle 
before  the  observation,  and  when  a  deflection  has  been 
produced  by  the  passage  of  a  current,  the  coil  is  rotated 
and  the  attempt  made  to  bring  it  once  more  parallel  to 
the  needle.  Every  movement  of  the  coil  produces  a 
further  displacement  of  the  needle  and  the  two  are  brought 
into  the  same  vertical  plane  only  by  careful  adjustment. 
When  the  coil  and  needle  are  parallel,  the  moment  of  the 
earth's  directive  force  is,  as  before,  mlH  sin  6,  but  the 
deflecting  influence  of  the  current,  acting  always  at  right 
angles,  is/;«/ 

Equating  f=Hs'mB. 

In  the  same  galvanometer, /is  always  a  function  of  C, 
and  hence  the  current  varies  as  the  sine  of  the  angle  of 
deflection.  Thus  knowing  the  deflection  that  a  current 
of  known  strength  produces 

C :  C'  :  :  sin  0  :  sin  0'. 
No  current  can   produce  a  deflection  of  more  than  90° 


NOTES  ON 


in  a  tangent  galvanometer  or  in  one  in  which  the  coil  is 
fixed.  In  a  sine  galvanometer,  however,  as  the  coil  is 
always  kept  parallel  to  the  needle  it  exerts  the  same  de- 
flecting force  in  all  positions.  If  now  the  current  is  of 
such  a  strength  that  equilibrium  is  attained  at  90°  from 
the  original  position  of  the  needle,  a  stronger  current 
would  deflect  it  still  farther,  and  it  would  be  impossible  to 
obtain  equilibrium. 

3.  Mirror  Galvanometer  (§  202). 

The  supposition  is  generally  made  that  with  the  mirror 
galvanometer  the  currents  vary  directly  as  the  scale  read- 
ings, but  this  is  true  only  within  limits.  The  needle  being 
small  and  the  coil  large,  the  current  is  proportional  to  the 
tangent  of  the  deflection,  but  as  the  deflection  is  read  by 
the  movement  of  a  spot  of  light  on  a  tangent  scale,  the 
current  would  be  proportional  to  the  reading,  if  it  did 
not  follow  from  the  laws  of  reflection  that  the  spot  of  light 
moved  over  twice  the  angle  that  the  mirror  moved. 
Calling  the  observed  deflections  d  and  d't  the  true  ratio  is 

C :  O  :  :  tan  -  .  tan  — . 

In  assuming  that  the  currents  are  proportional  to  the 
readings,  the  supposition  is 

C :  C'  :  :  r  :  r'  :  :  tan  d  :  tan  d' . 

If  tan  d  :  tan  d'  :  :  tan   — .  tan — ,   as  it   is  sensibly 

with  small  values  of  d  and  d't  the  deflections  may  be  taken 
to  be  proportional  to  the  strengths  of  the  currents.  In  the 
mirror  galvanometer,  therefore,  the  currents  are  propor- 
tional to  the  readings  if  the  deflections  are  small. 

4.  Differential  Galvanometer  (§  203). 
The  differential  galvanometer  is  used   not  to  measure 


ELECTRICITY   AND    MAGNETISM. 


currents,  but  to  indicate  that  two  are  either  equal  or  un- 
equal. As  the  greater  the  resistance  a  current  has  to 
flow  through,  the  smaller  the  current  is,  by  changing  the 
resistances  in  the  circuit  of  either  of  the  two  coils  of  the 
galvanometer  the  currents  may  ba  made  equal.  It  af- 
fords, therefore,  an  easy  method  of  comparing  resistances. 


Fig.  2. 

In  the  figure  it  is  seen  that  if  the  key  is  depressed  to  make 
contact  with  EE,  the  currents  pass  around  the  needle  in 
opposite  directions.  In  the  figure  one  current  is  farther 
from  the  needle  than  the  other,  but  great  care  is  taken  in 
making  the  instrument  to  have  the  coils  similarly  placed 
and  of  equal  resistance,  so  that  equal  currents  will  flow 
through  them,  producing  also  equal  but  opposite  effects 
on  the  magnet.  If  there  is  no  deflection,  the  currents  are 
equal,  or  C  —  C  ; 

E  E 


but  C  = 


and  C  = 


But  since  a  =  b,  x  =  ft,  if  there  is  no  deflection  of  the 
needle  when  the  key  is  pressed.  By  having  R  adjusta- 
ble, x  may  be  determined. 


NOTES   ON 


5.  Ballistic  Galvanometer  (§  204). 

When  a  current  is  of  very  short  duration,  it  may  be 
supposed  to  exert  an  impulsive  force  on  a  galvanometer 
needle,  especially  if  the  latter  is  heavy,  as  it  would  then, 
by  virtue  of  its  inertia,  fail  to  move  until  all  the  varying 
impulses  of  the  transient  current  had  been  given  it.  The 
effect  is  then,  practically,  that  produced  by  a  momentary 
impulse,  and  the  needle  will  move  with  a  velocity  pro- 
portional to  the  quantity  passing,  the  force  exerted  in  any 
case  varying  as  the  current  and  the  time  the  current 
lasts,  or  as  the  quantity  if  the  time  is  small,  and  will  vibrate 
through  a  certain  arc,  coming  to  rest,  and  then  under 
the  directive  action  of  the  earth's  magnetism  will  make  a 
return  oscillation  acquiring  the  same  velocity  as  that 
originally  given  by  the  current.  In  this  oscillation  the 
effective  force  tending  to  bring  it  to  rest  varies  as  the 
sine  of  the  angle  of  deflection,  as  it  does  also  in  the  case 
of  a  simple  pendulum,  and  the  relations  deduced  from  the 
latter  may  therefore  be  applied  without  perceptible  error 
to  the  vibrating  magnet. 

The  velocity  acquired  by  the  pendu- 
lum in  falling  through  the  arc  D  B  is 
equal  to  that  it  would  acquire  in  falling 
from  C  to  B ;  hence  from  the  laws  of 
falling  bodies 


v  =  V 
But  BC  =  AD  (i  -cos  6)=2  AD  sin2 1 0 


Fig-  3-  calling  AD,  I 


=  V  4  Ig  sin2  £  0 

0          • 


v  =  2  sn 


But  as  the  velocity  is,  proportional  in  the  galvanometer  to 


ELECTRICITY   AND   MAGNETISM.  7 

the  quantity  passing,  the  quantity  varies  as  the  sine  of 
half  the  angle  of  deflection.  If  the  deflection  is  noted 
when  a  known  quantity  is  discharged  through  a  ballistic 
galvanometer,  any  other  quantity  may  be  determined  by 
taking  the  ratio  of  the  sines  of  half  the  respective  deflec- 
tions. 


NOTES   ON 


II.  THEORY   OF   POTENTIAL. 

(Thompson's  Electricity,  pages   190-208.) 

6.  Potential  (§  237). 

An  explanation  of  the  term  potential  must  precede 
any  definition.  To  illustrate,  suppose  a  weight  of  a  pound 
be  moved  by  the  hand.  In  order  to  lift  it  to  a  higher 
level  the  muscles  have  to  be  called  on  to  do  work.  If  the 
pound  is  lifted  ten  feet,  it  is  only  by  the  expenditure  often 
foot  pounds  of  work.  If  after  being  lifted,  it  is  placed  on 
a  shelf,  the  work  done  on  it  is  evidently  in  the  form  of 
potential  energy,  and  may  be  recovered  if  the  weight  is 
allowed  to  fall,  when  it  will  do  ten  foot  pounds  of  work. 
Strictly  speaking,  the  weight  on  the  shelf  possesses  energy 
by  virtue  of  the  work  done  on  it,  but  if  this  work  had  not 
been  done  visibly,  the  weight  might  be  said  to  possess 
energy  by  virtue  of  its  mass  and  the  potential  or  height 
to  which  it  had  been  raised.  A  stone  on  the  top  of  a  prec- 
ipice is  thus  said  to  have  energy  by  virtue  of  its  potential, 
disregarding  th2  question  of  how  much  work  is  required 
to  lift  it  to  its  position.  Let  a  weight  of  two  pounds  be 
now  lifted  to  the  same  shelf,  at  a  height  or  potential  of 
ten  feet.  If  allowed  to  fall  it  would  do  twenty  foot 
pounds  of  work,  although  falling  from  the  same  height  or 
potential  as  did  the  other  weight  which  performed  only  ten 
foot  pounds  in  its  descent.  The  work  has  been  doubled,  al- 
though the  potential  is  the  same.  The  work  done,  either  on 
the  weight  in  lifting  it  to  the  shelf,  or  by  the  weight  in  falling, 
is  evidently  the  product  of  the  weight  into  the  potential. 


ELECTRICITY   AND    MAGNETISM.  9 

If  a  common  bar  magnet  be  taken  in  the  hand  and 
moved  near  a  powerful  fixed  magnet,  work  will  be  done. 
If  like  poles  are  near  each  other,  a  force  of  repulsion  is 
exerted,  and  the  muscles  are  called  upon  to  do  work  in 
bringing  the  magnet  nearer  to  the  fixed  magnet,  moving 
it  against  this  force.  If  the  bar  magnet  be  now  suspended 
so  as  to  be  unable  to  turn  end  for  end,  it  will  when  re- 
leased by  the  hand  fly  away  from  the  fixed  magnet  under 
the  influence  of  the  force  of  repulsion,  doing  work  in  its 
movement.  The  work  done  against  the  magnetic  forces 
in  bringing  the  bar  magnet  nearer  the  fixed  magnet  be- 
comes potential  energy,  and  is  available  as  kinetic  when- 
ever the  restraining  force  of  the  hand  is  removed.  Here, 
as  in  the  case  of  the  weight,  the  potential  energy  is 
derived  from  the  work  previously  done  in  moving  the 
magnet,  but  it  is  simpler  to  say  that  the  magnet  possesses 
energy  by  virtue  of  the  potential  to  which  it  is  raised. 
If  another  magnet  of  double  the  strength  be  moved, 
twice  as  much  work  will  have  to  be  expended  on  it  in 
bringing  it  to  the  position  occupied  by  the  first,  and  it 
will  have  twice  the  potential  energy.  As  it  has  been 
brought  to  the  same  potential  the  double  work  is  due  to 
its  being  of  twice  the  strength.  In  this  case,  therefore,  work 
is  the  product  of  strength  of  pole  and  magnetic  potential. 

Suppose  a  unit  quantity  of  positive  electricity  be  moved 
near  a  larger  quantity  also  positive.  A  force  of  repulsion 
exists  between  them,  and  if  they  be  brought  nearer 
together,  the  unit  quantity  will  if  released  fly  away  from 
the  larger  quantity,  doing  work.  It  evidently  possesses 
potential  energy,  or  does  work  by  virtue  of  its  potential. 
If  a  charge  of  two  units  be  brought  up  twice  the  work 
will  be  done,  and  there  will  be  twice  the  potential  energy. 
Here  work  is  evidently  the  product  of  quantity  of  elec- 
tricity and  electrostatic  potential. 


10  NOTES    ON 


7.  Difference  between  Work  and  Potential. 

As  in  each  of  the  three  cases  examined  the  work 
done  or  the  potential  energy  possessed  is  the  product  of 
potential  and  some  other  factor,  if  that  factor  be  known 
the  potential  may  be  obtained  by  measuring  the  work. 
Potential  is  therefore  measured  by  work,  but  is  not  work. 
In  several  places  in  "  Thompson's  Electricity  "  the  state- 
ment is  made  that  "  potential  is  the  work."  The  difference 
between  them  may  appear  more  clearly  from  a  recapitu- 
lation of  the  relations  already  traced. 

Work   done   in  lifting  weight 

Gravitation  Potential    =    TJ .  .    .  _.  . . .,  f — 

Weight  lifted. 

Magnetic   Potential      = 

Work  done  in  moving  magnet  pole 

Strength  of  pole. 
Electrostatic  Potential  = 

Work  done  in  moving  quantity  of  electricity 
Quantity  moved. 

Having  thus  traced  the  general  analogies,  in  further 
consideration  electrostatic  potential  alone  need  be  con- 
sidered. From  the  last  equation  it  is  seen  that  if  unit 
quantity  be  moved,  the  potential  is  numerically  equal  to 
the  work  done  in  moving  it.  As  the  work  done  in  moving 
from  zero  potential  is  the  measure  of  the  potential  energy 
acquired,  the  electrostatic  potential  at  a  point  equals 
the  potential  energy  possessed  by  unit  quantity  of  posi- 
tive electricity  at  that  point,  and  is  measured  by  the 
work  that  must  be  spent  in  bringing  unit  quantity  of 
positive  electricity  up  to  the  point  from  an  infinite  dis- 
tance. The  infinite  distance  enters  the  definition  from 
the  fact  that  there  the  potential  is  zero.  This  is  evident 


ELECTRICITY    AND    MAGNETISM.  II 

from  the  fact  that  potential  is  measured  by  work  ;  work 
is  the  product  of  force  into  distance,  over  which  it  acts, 

and  force  =  JQL  .     If  r  is  infinite,  the  force  is  zero,  and  no 

r* 

work  is  done  in  moving  unit  quantity.  „ 

8.  Positive  and  Negative  Work. 

As  in  any  case  in  which  a  positive  unit  is  repelled,  a 
negative  unit  is  attracted,  the  work  which  in  the  first  case 
is  necessary  to  move  the  unit  against  the  force  of  repulsion 
would  in  the  second  be  done  in  preventing  the  movement 
of  the  unit  under  the  forces  of  attraction.  The  two  cases 
are  evidently  diametrically  opposite,  and  the  work  done  is 
therefore  considered  as  positive  when  it  is  done  on  the 
the  unit,  negative  when  done  by  the  unit  moving  freely. 
It  becomes  necessary,  therefore,  to  specify  the  positive 
unit  in  the  definition,  that  the  nature  of  the  work  and  con' 
sequently  the  sign  of  the  potential  may  be  known. 

9.  Positive  Electricity  always  flows  from  a  High  to  a  Low 
Potential. 

Potential  energy  always  tends  to  run  down  to  a  mini- 
mum. A  weight  acted  on  by  gravity  will  fall  to  the 
earth  if  not  prevented  ;  a  magnet  pole  placed  near  another 
similar  pole  possesses  potential  energy  and  tends  to  move 
away  into  a  position  in  which  its  potential  energy  is 
less  ;  a  unit  of  electricity  placed  near  a  similar  quantity 
is  likewise  repelled  and  moves  so  as  to  decrease  its  poten- 
tial. In  any  electrified  region  the  relative  potential  is 
therefore  indicated  by  the  direction  in  which  a  unit  of 
positive  electricity  tends  to  move,  and  the  distribution  of 
potential  may  be  examined  by  conceiving  a  positive  unit 
to  be  moved  throughout  the  neighborhood  of  the  electrified 


12  NOTES   ON 


bodies,  noting  whether  it  is  necessary  to  do  work  to  move 
it  or  to  restrain  its  movement.  Let  this  positive  unit  be 
approached  to  a  quantity  of  positive  electricity.  Work 
must  be  done  to  move  it,  and  if  left  free  it  will  fly  away, 
moving  to  decrease  its  potential.  It  has  evidently  been 
moved  into  a  region  of  higher  potential.  If  approached 
to  a  negative  unit,  work  must  be  done  to  restrain  its 
movement.  If  free  to  move  it  will  move  toward  the 
negative  unit,  and  is  moving  into  a  region  of  lower 
potential.  Generally  speaking,  a  body  positively  electri- 
fied is  at  a  positive  potential,  and  one  negatively  electri- 
fied at  a  negative,  but  there  are  many  exceptions. 

Suppose  a  cylinder   B  to  be   unelectrified  and   to   be 
connected  with  the  earth  by  a  wire.     There  is  no  flow  of 

electricity  in  the  wire,  and 
B  is  therefore  at  the  same 
potential  as  the  earth,  as 
electricity  tends  to  move 
Fig.  4.  toward  a  lower  potential, 

and  the  fact  of  there  being  no  movement  shows  there  is 
no  difference  of  potential.  Let  a  positively  electrified  ball 
A  be  approached,  and  B  becomes  electrified  by  induction 
as  in  the  figure.  If  again  connected  with  the  earth,  a  flow 
of  positive  electricity  takes  place  from  the  cylinder  to  the 
earth,  showing  that  the  potential  of  B  had  been  raised  by 
the  approach  of  A.  Before  being  connected  with  the 
earth  the  second  time  it  was  therefore  at  a  positive  poten- 
tial but  was  negatively  electrified  at  one  end.  When  in 
contact  with  the  earth,  with  A  as  in  the  figure,  it  is 
negatively  electrified,  but  at  zero  potential. 

10.  Units  of  Potential  and  Work. 

Potential  is  measured  by  work,  and  the  units  of  poten- 
tial are  numerically  equal  to  the  units  of  work.     Work  is 


ELECTRICITY  AND   MAGNETISM.  13 

defined  as  force  acting  through  distance,  and  the  C.  G.  S. 
unit  of  work  called  the  erg  is  the  work  done  in  opposing 
the  force  of  one  dyne  through  the  distance  of  one  centi- 
metre. It  may  be  defined  after  the  analogy  of  foot-pounds 
as  a  dyne-centimetre.  If,  therefore,  the  work  necessary 
to  move  a  unit  quantity  of  electricity,  or  the  work  a  unit 
quantity  does  in  moving,  is  measured  in  ergs,  it  numeri- 
cally equals  the  difference  of  potential  through  which  the 
unit  moves. 

ii.  Electrostatic  Potential  (§  238). 

We  can  now  derive  the  general  formula  for  electrostatic 
potential.     From  the  definitions  of  potential  and  work  — 

.00 

Work  =r        fdrt  where  r  is  distance  j 


.0 

fdrt 
J  r 


but  /  =  9l       and  Work  =  -  dr. 

*  r 

If  q  is  unity,  work  measures  potential,  hence,  denoting 
potential  by  V, 


The  potential  at  any  point  due  to  a  quantity  q  is,  there- 
fore, numerically  equal  to  the  quantity  divided  by  the  dis- 
tance in  centimetres.  If  other  quantities  q' ' ,  q" ,  etc.,  were 

near,  the  potential  due  to  them  would  be  —  »  —  ,  etc.  The 
potential  due  to  the  whole  system  is  then 

v=<-  +  t  +  C  =  2i. 

r        r        r'  r 


NOTES  ON 


If  either  q,  q  or  q"  is  negative,  —  »  —  or  —  will  be  nega- 
tive, and  must  be  given  its  proper  sign  in  the  summation. 


12.  Zero  Potential  (§  239). 

Although  as  shown,  the  theoretical  zero  potential  exists 
at  an  infinite  distance,  the  potential  of  the  earth  at  the 
place  is  the  practical  zero.  All  electrical  manifestations 
are  dependent  on  a  difference  of  potential,  and  the  abso- 
lute potential  is  never  needed. 

13.  Difference  of  Potentials  (§  240). 

Potential  being  measured  by  work  done,  the  difference 
of  potential  between  two  points  is  numerically  equal  to  the 
number  of  ergs  required  to  move  a  positive  unit  from  one 
point  to  the  other.  It  is  immaterial  what  path  be  followed, 
as  if  all  the  work  done,  both  positive  and  negative,  be 
summed  up,  it  will  be  equal  to  that  done  in  moving  in 
a  direct  line  between  the  points.  Two  quantities  of  elec- 
tricity at  different  potentials  may  be  compared  to  two 
ponds  of  water  at  different  levels.  If  the  ponds  are  con- 
nected by  a  pipe,  the  water  in  the  upper  will  by  virtue  of 
its  height  possess  potential  energy  and  will  run  down  into 
the  lower.  If  no  current  of  water  flowed  in  the  pipe  it 
would  indicate  that  the  ponds  were  at  the  same  level.  If 
no  other  means  of  measuring  the  difference  of  level  were 
available,  it  could  be  done  by  measuring  in  foot-pounds 
the  work  done  by  one  pound  of  water  in  flowing  through 
the  pipe.  Similarly  electricity  flows  from  a  body  electri- 
fied to  a  high  potential  to  one  at  a  lower  connected  with 
it,  and  the  difference  of  potential  is  measured  by  the  work 
in  ergs  done  by  unit  quantity  in  flowing  from  one  to  the 
other. 


ELECTRICITY  AND   MAGNETISM.  15 

14.  Electric  Force  (§  241). 
When_/  is  force  exerted  on  unit  quantity 

V=  \fdrtwdV=fdr    .:/  =  —  . 

dV 
But  -j—  is  the  rate  of  change  of  potential,  hence   the 

average  electric  force  between  two  points  at  different  po- 
tentials is  measured  by  the  rate  of  change  of  potential  per 
centimetre. 

15.  Law  of  Inverse  Squares  (§  245,  Fig.  98). 

Coulomb's  observations,  without  proving  exactly  that 
the  1  >w  of  inverse  squares  applied  to  electric  force,  so 
nearly  proved  it  as  to  lead  one  to  think  that  more  careful 
experimentation,  were  such  possible,  would  demonstrate 
the  exactness  of  the  law.  Assume,  therefore,  the  law  and 
trace  the  results. 

Let  p  be  the  electric  density,  or  the  amount  of  electricity 
per  square  centimetre  of  surface.  On  a  sphere  removed 
from  other  conductors  the  density  is  uniform,  and  the 
quantity  on  two  surfaces  varies  as  the  area  of  the  surfaces. 
The  quantity  on  the  surface  AB  (Fig.  98,  "  Thompson  ") 
is  p  x  Area  AB  —  pA. 

The  quantity  on  CD  is  p  x  Area  CD  =  pC. 

Assuming  that  electric  force  varies  inversely  as  the 
square  of  the  distance,  the  force  exerted  on  a  unit  of  elec- 
tricity at  the  point  P  inside  the  sphere  by  the  quantity  on 
ABis 


The  force  exerted  on  the  same  unit  by  the  quantity  on 
CD  is 


1 6  NOTES   ON 


Cf 

Since  the  tangents  drawn  to  the  sphere  are  equally  in- 
clined to  JSC 

C  x  BP* 

A\C\\BP»\  CP* ,      orA  =  —= ' 

LP* 

substituting  above/  =  /'. 

If  the  sphere  be  cut  up  into  small  cones  2f  =  2f  ;  or  in 
other  words,  if  electric  force  varies  inversely  as  the  square 
of  the  distance,  there  should  be  no  resultant  force  on  the 
inside  of  a  closed  conductor.  If  it  follows  any  other  law 
/cannot  equal/'.  The  most  careful  experiments  fail  to 
detect  any  force  existing,  and  in  corroborating  the  result 
of  the  above  demonstration,  confirm  the  hypothesis  made 
that  electric  force  follows  the  law  of  inverse  squares. 

1 6.  Capacity  (3  246). 

The  capacity  of  a  conductor  is  by  the  definition  given 
a  fixed  quantity,  while  "the  amount  of  electricity  the  con- 
ductor can  hold,"  the  definition  as  generally  given  by  be- 
ginners, is  variable,  depending  on  the  potential  as  well  as 
on  the  capacity.  An  illustration  may  make  the  distinction 
clearer.  If  a  jar  has  a  volume  of  one  litre,  its  capacity  is 
a  litre,  and  it  will  hold  a  litre  of  air  at  atmospheric  press- 
ure. If  the  pressure  be  doubled,  however,  the  quantity 
of  air  in  the  jar  is  also  doubled,  although  the  capacity  is 
the  same.  "The  amount"  the  jar  "will  hold"  is  evi- 
dently determined  by  the  capacity  and  pressure.  If  the 
pressure  be  unity,  that  of  one  atmosphere,  the  capacity  is 
then  the  amount  the  jar  actually  holds,  but  quantity  and 
capacity  are  under  other  conclitious  different.  With  ref- 
erence to  electricity,  the  capacity  is  similarly  the  charge 


ELECTRICITY   AND    MAGNETISM.  I? 

the  conductor  will  hold  at  unit  potential,  or  the  charge 
which  will  raise. the  potential  to  unity,  and  the  actual 
charge  in  a  conductor  is  the  product  of  the  capacity  and 
potential. 

17.  Unit  of  Capacity  (§  247). 

In  an  electrified  sphere,  as  the  surface  is  an  equipoten- 
tial  surface,  the  charge  may  be  considered  as  concentrated 

at  the  centre  and  the  potential  at  the  surface  is  -  •     As 

the  capacity  is  equal  to  the  quantity  divided  by  the  poten- 
tial 

Capacity  =  — -  =  r. 


A  sphere  of  one  centimetre  radius  is,  therefore,  of  unit 
capacity. 

18.  Electric  Force  exerted  by  a  Charged  Plate  (§  252). 

Let  a  be  the  radius  of  the  plate  and  r  the  radius  of  the 
ring  x,  x,'  x."     p  is  the 
density,  or  the   charge 
per  unit  of  area. 

The  quantity  on  any 
small  circular  element 
is  p  (27trdr).  The  force 
exerted  by  this  quan- 
tity on  a  unit  at  O  is 

p(2itrdr) 


and  the  force  acting  nor- 
mal to  the  plate  is 


Fig.  5 


1 8  NOTES  ON 


2-itrdr  h       \  h 

P     [    vv-        .,     •    -  =  -  )  ,  Since  COS  0  = 

* 


The  total  force  exerted  by  the  plate  in  a  direction  normal 
to  its  surface  is,  therefore 

*  iitrdr  h  ,    C  a   2  rdr 

'  = 


Integrating,  =  —  27tph  (h 


+  r2)i    I 


=  27tph  h 

' 27tp  ,    


—  27tp  (I    —  COSO'). 

If  O  is  very  near  the  plate,  or  if  the  plate  is  very  large, 
6'  =  90°,  and  the  electric  force  of  a  charged  plate  on  a  unit 
of  electricity  very  near  it  is  2itp. 

Care  must  be  taken  not  to  confuse  this  with  the  force 
exerted  by  a  sphere  as  deduced  in  g  251.  If  a  plate  is 
charged  with  positive  electricity,  and  a  positive  unit  is 
placed  very  near  it  on  each  side,  the  force  will  be  one  of 
repulsion  in  each  case,  but  if  one  unit  is  repelled  upward 
the  other  tends  to  move  downward,  and  if  one  force  is 
2it p,  the  other  must  be  —  2rtp.  The  force  changes,  there- 
fore, by  47tp  in  passing  through  any  charged  surface. 

19.  Dimensions  of  Units  (§  258). 

An  important  use  of  the  dimensional  equations  is  in  the 
conversion  of  units  based  on  one  system  of  fundamental 
units  of  mass,  length  and  time  to  others  based  on  different 
fundamental  units.  The  French  use  units  based  on  the 
metrical  system,  and  although  the  centimetre-gramme- 
second,  or  C.  G.  S.  system,  is  now  almost  universally  used 


ELECTRICITY   AND   MAGNETISM.  19 

in  electrical  work,  there  are  many  observations  made  and 
recorded  in  which  other  units  are  used.  In  electrical 
work  the  English  have  heretofore  used  the  foot-grain- 
second  system,  and  it  is  still  used  in  some  government 
observatories.  It  is  a  matter  of  the  highest  importance, 
therefore,  that  the  method  of  converting  values  expressed 
in  one  system  to  corresponding  values  in  another  should 
be  thoroughly  understood.  As  the  ratio  of  the  centi- 
metre to  the  foot  is  that  of  I  to  30.48,  it  is  evident  that 
the  ratio  of  the  units  of  length  in  the  C.  G.  S.  and  foot- 
grain-second  systems  is  the  same.  The  units  of  area  in 
the  two  systems  are  the  square  centimetre  and  the 
square  foot,  and  no  one  would  think  of  saying  that  this 

ratio  was  the  same    as  the  preceding -,  but  rather 

— -1 —  .  Similarly  the  ratio  between  the  units  of  volume 
(3048)' 

i  i3 

is  not but  . 5T7-.      This    is    exactly   the    relation 

30.48         (30.48)' 

shown  by  the  dimensions  of  area  and  volume  on  page 
211,  they  being  respectively  Z,2  and  D  .  In  simple  cases 
like  the  above  the  change  is  easily  made,  but  in  others, 
where  the  dimensions  of  the  unit  are  more  complex  and 
the  unit  itself  an  unfamiliar  one,  the  dimensions  must  be 
used  to  calculate  the  ratio.  As  an  illustration,  let  it  be 
required  to  express  in  units  of  potential  based  on  the  foot, 
grain  and  second,  the  difference  of  electrostatic  potential 
expressed  by  2.7  C.  G.  S.  units  of  potential. 
The  dimensions  of  electrostatic  potential  are  M^  L£  T~  , 

The  C  G.  S.  unit  /;;Ai//\i 

=  \M)    \L) 


then 


Foot-Grain-Sec,  unit 


_ 

30.48; 


=  >7II6 


20  NOTES   ON 


.'.    I  C.  G.  S.  unit  =  .7116  Foot-grain-second  unit 
2.  7  C.  G.  S,  =.   1.92  Foot-grain  sec. 

The  ratios  between  the  different  units  in  the  two  systems 
are  given  in  Note  40. 

20.  Attracted  Disc  Electrometers  (§  261,  p.  215). 

Let  the  difference  of  potential  between  two  plates  be  V, 
and  the  distance  apart  be  D.     By  Note  14  the  average 

y 
electric  force  between  the  plates  is  —  .    As  proved  in  Note 

1  8,  the  electric  force  changes  by  ^itp  in  passing  through  a 
surface,  and   being  zero  in    the    conductor   is    therefore 

y 
47tp  just  outside.    Equating,  /o  =  -  —.      The  density   on 


each  plate  being  p,  the  attraction  exerted  by  the  lower  plate 
on  a  unit  of  electricity  on  the  upper  one  is  2jrp  when  the 
plates  are  near  each  other.  The  upper  plate  contains, 
however,  Sp  units  and  the  total  attraction  is  iitp  x  Sp  = 


21.  Absolute  Electrometer  (p.  216). 

Sir  William  Thompson's  absolute  electrometer,  so  named 
from  giving  the  potential  in  absolute  units,  is  an  attracted 
disc  electrometer. 

The  disc  C  (see  Figure  TOO,  "  Thompson")  is  held  in  place 
by  springs,  instead  of  a  counterpoise  as  shown,  and  is  in 
metallic  connection  with  B.  When  no  part  of  the  appa- 
ratus is  electrified,  small  weights  are  placed  on  C,  to  bring 


ELECTRICITY   AND    MAGNETISM.  21 

it  into  a  standard  position  such  that  a  small  hair  attached 
to  it  is  seen  midway  between  two  dots,  as  shown  in  the 
figure.  The  weights  are  then  removed  and  B  and  C  con- 
nected to  one  of  the  bodies  whose  difference  of  potential 
is  required  and  A  to  the  other.  The  electric  force  of 
attraction  between  the  two  plates  will  act  to  lower  C,  but 
as  the  accuracy  of  the  instrument  depends  on  its  being  in 
the  plane  of  B,  the  plate  A  is  moved  up  or  down  until 
the  force  of  attraction  is  such  as  to  bring  the  movable 
plate  into  this  standard  position,  which  is  known  by  see- 
ing the  hair  again  midway  between  the  dots.  It  is  now 
under  the  attraction  of  the  electrical  forces,  in  exactly  the 
same  position  as  when  acted  upon  by  the  weights,  and 
the  two  forces  are  therefore  equal.  Substituting,  there- 
fore, for  .Fin  the  formula  of  Note  20,  981  times  the  weight 
in  grammes  required  to  bring  C  into  the  standard  position, 
and  for  D  the  distance  in  centimetres  between  A  and  Ct 
all  quantities  in  the  equation  are  known  and  the  difference 
of  potential  may  be  calculated. 

Another  method  is  more  common,  dispensing  with  the 
use  of  weights.  If  the  difference  of  potential  between  two 
bodies  P  and  P'  is  required,  one  of  the  bodies  is  connected 
to  A,  and  B  and  C  are  then  electrified  to  a  high  potential. 
The  plate  A  is  then  moved  up  or  down  until  the  plate  C 
comes  into  the  standard  position,  the  hair  showing  mid- 
way between  the  dots,  and  the  distance  D  of  A  from  Cis 
noted.  Then 


Potential  of  B  -  Potential  of  P  =  D  y 

The  plate  A  is  next  disconnected  from  the  first  body 
and  connected  with  the  second.  As  the  difference  of 
potential  between  A  and  Chas  now  been  changed,  the 
force  acting  between  the  plates  is  different  and  C  is  no 


22  NOTES  ON 


longer  in  the  standard  position.  It  is  brought  there  by 
raising  or  lowering  A,  and  when  adjusted  the  distance  D' 
between  the  plates  is  noted  r 

Then  Potential  of  B—  Potential  of  P'  =  D  'i/8^ 

V     S. 

Subtracting  this  from  the  former, 

Difference  of  Potential  between  P  and  P'  —  (D—D')  x 
constant  of  instrument. 

It  is  of  course  necessary  that  the  potential  of  B  should 
be  the  same  in  both  cases.  This  is  verified  by  a  separate 
attracted  disc,  which  is  in  each  case  electrified  until  its 
attraction  for  another  disc  at  a  fixed  distance  brings  it 
into  a  standard  position.  The  absolute  potential  of  B  is 
immaterial,  the  only  requirement  being  that  it  should  be 
the  same  in  each  case.  If  the  absolute  potential  of  P  is 
wished,  connect  A  first  to  P  and  then  to  earth. 


ELECTRICITY   AND    MAGNETISM.  23 


III.  THEORY  OF   MAGNETIC  POTENTIAL. 

(Thompson's  Electricity,  pages  265-278.) 

22.  Magnetic  Field. 

Any  region  throughout  which  forces  act  is  called  a 
"  field,"  but  the  term  is  more  frequently  used  in  connec- 
tion with  magnetic  than  with  other  forces.  A  magnetic 
field  is,  therefore,  a  region  in  which  magnetic  effects  are 
produced.  Any  movement  of  a  magnet  pole  can  take 
place  only  in  a  magnetic  field,  and  the  term  is  of  use,  as 
it  disregards  all  ideas  of  how  the  field  is  caused,  and  con- 
siders only  the  forces  and  the  direction  in  which  they  act. 
If  at  any  point  a  line  is  drawn  indicating  the  direction  of  the 
force  at  that  point,  it  is  called  a  line  of  force.  This  direc- 
tion is  that  shown  by  a  magnet  placed  at  the  point,  and 
may  therefore  be  easily  determined  by  experiment  ;  but  as 
any  representation  of  a  magnetic  field  must  present  the 
whole  field  at  once,  the  determination  of  the  position  of 
the  lines  of  force  by  this  process  would  be  tedious.  The 
reasoning  in  \  126  leads  to  an  easier  though  less  accurate 
method,  but  one  of  great  utility  in  enabling  clear  concep- 
tions to  be  formed.  If  a  magnet  is  covered  by  a  sheet  of 
paper  and  iron  filings  are  sprinkled  over  the  paper  they 
will  on  being  gently  tapped  arrange  themselves  in  curves 
passing  from  pole  to  pole.  From  the  definition  of  lines  of 
force,  these  curves  must  be  the  lines  of  force  in  the  plane 
of  the  paper,  and  the  mind  has  only  to  conceive  the  space 
above  and  below  the  magnet  to  be  similarly  filled  to  gain 
a  clear  idea  of  the  field.  It  is  necessary,  however,  to 


24  NOTES  ON 


know  not  only  the  direction  of  the  force  at  any  point,  but 
also  its  strength,  and  a  correct  plotting  of  the  field  must 
furnish  this.  Maxwell  has  shown  that  if  in  any  part  of 
their  course,  the  number  of  lines  of  force  passing  through 
unit  area  of  a  perpendicular  plane  is  proportional  to  the 
strength  of  the  force  there,  the  number  passing  through 
unit  area  in  any  other  part  of  the  field  is  in  the  same  pro- 
portion to  the  strength  in  that  part.  The  closeness  of  the 
lines  of  force  is  therefore  a  measure  of  the  strength  of  the 
forces  of  the  field,  or,  as  more  commonly  expressed,  of  the 
intensity  of  the  field.  By  drawing  the  lines  of  force 
therefore  in  this  way,  the  strength  and  direction  of  the 
forces  in  all  parts  of  the  field  are  indicated.  As  a  south 
pole  moves  always  in  the  opposite  direction  to  that  in 
which  a  north  pole  moves,  it  is  necessary  in  order  to 
establish  the  direction  of  the  force  to  consider  the  nature 
of  the  pole  acted  upon.  All  investigations  in  magnetism 
are  made  by  considering  a  north  pole  free  to  move,  and 
the  positive  direction  of  the  lines  of  force  is  therefore 
that  in  which  a  free  north  pole  moves.  This  definition 
is  of  great  importance  in  many  of  the  demonstrations 
given  later. 

23.  Mapping  a  Field  by  Lines  of  Force. 

A  magnetic  field  is  of  unit  intensity  when  unit  pole  is 
acted  upon  by  a  force  of  one  dyne.  As  by  definition 
unit  pole  acts  on  an  equal  and  similar  pole  at  a  distance 
of  one  centimetre  with  a  force  of  one  dyne,  it  follows 
that  unit  pole  causes  unit  field  at  unit  distance.  As  in- 
tensity of  field  is  measured  by  the  force  acting  on  unit 
pole,  unit  field  exists  at  a  greater  distance  from  a  more 
powerful  pole.  It  is,  therefore,  unnecessary  to  consider 
the  question  of  distance  from  the  pole  producing  the  field, 
but  simply  bear  in  mind  that  the  intensity  of  the  field  at 


ELECTRICITY   AND   MAGNETISM.  25 

any  point  is  measured  by  the  force  in  dynes  acting  on  a 
unit  pole  at  that  point.  If  the  pole  be  of  a  strength  m, 
the  force  with  which  it  is  attracted  or  repelled  is  ;//  times 
that  experienced  by  unit  pole,  or 


Force  acting:  on  pole 

Intensity  of  field  =  —  =  —          -^~  —  f  -  • 
Strength  of  pole 

The  value  of  H,  or  the  strength  of  field,  is  given  numer- 
ically. Thus  the  horizontal  force  of  the  earth's  magnetism 
at  London  being  .18,  a  pole  of  unit  strength  is  impelled  to 
move  in  a  horizontal  plane  by  a  force  of  .18  dynes.  A 
pole  of  strength  100  would  be  acted  on  by  a  force  of 
1  8  dynes  in  a  horizontal  direction,  or  by  a  force  of  47 
dynes  in  the  line  of  dip.  To  represent  the  field  graphic- 
ally, recourse  is  had  to  Maxwell's  demonstration  of  the 
fact  that  the  number  of  lines  cutting  unit  area  in  different 
points  of  the  field  is  proportional  to  the  intensity  at  those 
points,  and  the  numerical  value  of  //"is  interpreted  as  the 
number  of  lines  per  square  centimetre  of  a  surface  per- 
pendicular to  the  direction  of  the  lines.  Thus  at  London 
the  earth's  horizontal  field  would  be  represented  by  draw- 
ing horizontal  lines  of  force  in  the  magnetic  meridian, 
equidistant  and  so  spaced  that  they  cut  a  vertical  east 
and  west  plane  at  the  rate  of  .18  per  square  centimetre 
or  of  one  line  to  every  5.56  +  square  centimetres.  The 
positive  direction  is  toward  the  north.  The  total  field  at 
London  would  be  represented  by  lines  of  force  in  the  di- 
rection of  the  dipping  needle,  equidistant  and  spaced  so 
as  to  cut  a  perpendicular  plane  at  the  rate  of  .47  per 
square  centimetre,  or  one  to  every  2.13  square  centimeters. 
These  lines  projected  intersect  at  the  magnetic  pole,  and 
are,  therefore,  sensibly  parallel  within  ordinary  limits,  in 
which  case  the  field  is  said  to  be  uniform. 


26  NOTES   ON 


24.  Equipotential  Surfaces, 

Being  surfaces  in  which  no  work  is  done  in  moving  a 
unit  pole,  are  necessarily  perpendicular  to  the  lines  of 
force.  If  not,  some  component  of  the  force  would  act,  and 
work  would  be  done  in  moving  against  it.  Knowing  the 
direction  of  the  lines  of  force,  the  equipotential  surfaces 
can  be  readily  drawn  by  cutting  all  the  lines  at  right 
angles.  Like  lines  of  force  they  may  be  drawn  in  any 
number  required,  but  it  is  customary  to  have  them  rep- 
resent unit  difference  of  potential,  and  this  requires  that 
they  should  be  so  far  apart  that  an  erg  of  work  is  done 
in  moving  a  unit  pole  from  one  to  the  other.  The  dis- 
tance may  be  readily  calculated. 

Work  =  Hx, 
but  by  definition  work  is  unity 

I 

•          y     — .     . L     • 

H  ' 

or  the  distance  of  the  equipotential  surfaces  is  inversely  as 
the  intensity  of  the  field.  The  field  may,  therefore,  be 
represented  in  this  way  as  accurately  as  by  the  lines  of 
force.  Taking  the  case  already  considered,  the  earth's 
horizontal  field  at  London  would  be  represented  by  verti- 
cal surfaces,  sensibly  planes,  extending  east  and  west  and 
5.56  +  centimetres  apart. 

25.     Lines  of  Force  due  to  a  Single  Pole. 

In  the  case  of  a  free  pole  of  strength  in,  the  number  of 
lines  of  force  is  determined  by  the  fact  that  there  are  ;;/ 
lines  intersecting  every  square  centimetre  at  unit  dis- 
tance, or  that  there  are  m  lines  cutting  every  square  centi- 
metre of  a  sphere  of  unit  radius.  The  surface  of  this 
sphere  being  471  there  are  in  all  ^itm  lines  of  force  radial- 


ELECTRICITY  AND   MAGNETISM.  27 

ing  equally  in  all  directions.  The  equipotential  surfaces 
are  spheres,  and  the  radii  may  be  found  from  the  for- 
mula for  magnetic  potential,  V— — .which  may  be  here 

assumed  to  be  correct.     By  substituting  values  for  Fdif- 

f    .       ,          .          .    f  mm 

fenng  by  unity,  r  is  found  to  be  successively  ;;/,  —  »    —  » 

— ,  etc.  As  a  free  pole  can  never  exist,  but  is  always  as- 
4 

sociated  with  another  of  equal  but  opposite  polarity,  any 
actual  field  is  much  more  complex,  but  the  cases  given 
will  illustrate  the  application  of  the  principles  traced, 
and  will  give  clear  ideas  of  the  conventions  made  which 
underly  further  investigation. 

It  is  possible  without  changing  the  number  of  lines  of 
force  to  change  their  distribution  in  the  field,  and  as  it  is 
frequently  desirable  to  intensify  a  certain  part  of  the  field, 
the  method  of  doing  so  becomes  of  importance.  A  com- 
parison of  Figures  52  and  53  "  Thompson's  Electricity," 
shows  that  the  distribution  may  be  greatly  changed  by 
an  arrangement  of  magnet  poles  in  the  field,  and  as  iron 
near  a  magnet  becomes  magnetized  by  induction,  the 
field  is  similarly  affected  by  the  introduction  of  iron.  In 
this  case  the  iron  seems  to  gather  the  lines  of  force  in  the 
vicinity,  causing  a  great  number  to  pass  through  its  sub- 
stance. Iron  placed  near  a  magnet  pole  becomes  mag- 
netized, so  that  dissimilar  poles  are  adjacent,  producing 
the  state  of  affairs  shown  in  Fig.  52.  Jenkin  compares 
this  peculiarity  of  iron  in  concentrating  the  lines  of  force 
to  that  of  a  lens  in  converging  rays  of  light.  It  is  likewise 
possible  to  screen  any  part  of  a  magnetic  field  from  in- 
duction, by  inclosing  it  in  an  iron  shell.  It  may  be  easily 
demonstrated  by  experiment  that  if  an  iron  ring  be  placed 
between  the  poles  of  a  horse-shoe  magnet,  no  lines  of 


28 


NOTES   ON 


force  pass  through  the  interior  of  the  ring,  but  entering 
at  one  side  pass  through  the  metal  of  the  ring  issuing  on 
the  opposite  side.  A  magnet  inside  an  iron  sphere  is  in- 
dependent of  all  outside  influences.  By  the  use  of  iron 
it  is,  therefore,  possible  either  to  concentrate  the  lines  of 
force,  or  to  divert  them  entirely  from  any  desired  part  of 
the  field. 

26.  Lines  of  Force  due  to  a  Current. 

As  shown  in  g  191,  the  lines  of  force  due  to  a  current 
are  circles  perpendicular  to  the  current  and  having  it  for 
a  centre.  If  the  conductor  is  straight,  the  circles  are  all 
in  parallel  planes,  and  the  equipotential  surfaces  are 
planes  radiating  from  the  conductor  and  each  containing 
it.  The  number  of  these  planes  is  such  that  an  erg  is 
required  to  move  unit  pole  from  one  to  the  other.  As 
shown  below,  the  intensity  of  the  field  at  unit  distance  is 
2(7.  A  pole  moving  in  a  circle  of  unit  radius  having  the 
conductor  for  a  centre  passes  over  a  distance  of  2?r  against 
a  force  of  2C,  doing  4?r(7  ergs 
of  work.  The  number  of 
equipotential  surfaces  is,  there- 
fore, 4-TtC.  After  the  pole  had 
made  one  revolution  it  would 
reach  the  equipotential  surface 
from  which  it  started,  but  hav- 
ing done  4/rC  ergs  in  its  revolu- 
tion the  numerical  value  of  the 
surface  would  now  be  471- C  more 
than  before.  It  is  impossible, 
A  therefore,  to  give  an  absolute 
value  to  an  equipotential  sur- 
.  6-  face  due  to  a  current. 

Let  LL  be  a  portion  of  an  infinite  rectilinear  current,  and  let 


ELECTRICITY  AND   MAGNETISM.  29 

BC  be  the  force  exerted  by  an  element  of  length,  dx>  of  this 
current  =  Cdx.     Place  a  unit  pole  at  A. 
The  force  exerted  by  the  element  dx  at  A  is 
_    Cdx  cos  Q  ^ 

Zg* 

If  <9^  =  i,   ~A&  =  i  +  x1 
i 


Cdx. 


Vi  +  x-  _        Cdx 
ar   = — t        — 


TT 

The  total  force  =  F  —  C  \  T~a  =  c  T      cos  6^9- 

•Loot1  +  *)»          J'i* 


Integrating,  ^=Csin0  —  zC  =  //. 


—  zC  = 


To  find  the  intensity  at  a  distance  r  from  the  conductor  we 
have  from  Note  24  that  the  intensity  is  the  reciprocal  of  the  dis- 
tance between  two  equipotential  surfaces.  There  being  4rtC 
surfaces  cutting  a  circumference  of  2rtr,  the  distance  between 


them  is  —  -  and  hence 
26? 


27.  Magnetic  Potential. 

Magnetic  potential  has  already  been  alluded  to  in  Note 
6.  The  conception  is  strictly  analogous  to  that  of  elec- 
trostatic potential,  and  the  demonstration  given  for  the 
formula  for  electrostatic  potential  in  Note  n,  is  applica- 
ble to  magnetic  if  for  q  a  quantity  of  electricity,  is  substi- 
tuted m  a  strength  of  magnet  pole.  The  same  reasoning 

leads  to  the  formula  of  V  •=.  2  —  • 

r 


3°  NOTES  ON 


Magnetic  potential  at  a  point  equals  the  potential 
energy  possessed  by  a  unit  north  pole  at  that  point, 
and  is  measured  by  the  work  in  ergs  done  in  bringing 
a  unit  north  pole  from  infinity  to  that  point. 

Zero  of  magnetic  potential  exists  at  an  infinite  distance 
from  all  mag-nets. 

dV 
Magnetic  force  is  — ,  or  the  rate  of  change  of  potential 

per  unit  of  distance  as  in  Note  14. 

The  difference  of  magnetic  potential  between  two 
points  is  measured  by  the  work  clone  in  moving  a  unit 
north  pole  from  one  to  the  other.  Wherever  work  has  to 
be  done  in  moving  a  north  pole,  it  would  be  done  in  re- 
sisting the  motion  of  a  south  pole.  In  all  investigations  of 
magnetic  potential,  force  or  work,  a  unit  north  pole  must 
always  be  considered. 

28.  Tubes  of  Force. 

The  conception  of  tubes  of  force  is  frequently  of  utility. 
The  lines  of  force  radiating  from  a  pole  may  be  regarded 
as  forming  cones,  and  any  section  through  a  cone  would 
cut  all  the  lines  of  force.  But  as  the  number  of  lines  of 
force  is  proportional  to  the  intensity,  the  force  on  all  cross 
sections  of  the  cone  is  therefore  the  same.  By  conceiv- 
ing the  magnetic  force  to  be  equal  throughout  the  cone, 
existing  between,  as  well  as  along  the  lines  of  force,  a 
more  accurate  idea  of  the  field  is  attained.  It  is  easy  to 
imagine  a  field  of  so  slight  intensity  that  a  square  centi- 
metre would  not  have  any  lines  of  force  passing  through 
it.  The  example  of  the  earth's  horizontal  field  at  London, 
already  referred  to,  is  a  case  in  ooint.  One  line  of  force 
passes  through  every  five  units  of  area,  but  the  magnetic 
forces  are  felt  just  as  strongly  on  the  four  units  through 
which  the  line  does  not  pass  as  on  that  which  it  cuts.  By 


ELECTRICITY   AND    MAGNETISM.  31 

thinking,  therefore,  of  the  lines  of  force  as  indicating  only 
the  direction  and  strength  of  the  forces  which  act  between 
them  as  well  as  in  them,  this  difficulty  is  overcome. 

29.  Intensity  of  Magnetization. 

If  a  bar  magnet  be  broken  in  half,  instead  of  obtaining 
one  piece  of  nortli  and  the  other  of  south  polarity,  each  is 
found  to  possess  both  and  to  be  a  perfect  magnet.  How- 
ever far  the  subdivision  be  carried,  the  result  is  the 
same,  and  the  ordinary  explanation  is  that  the  magnet  is 
an  aggregation  of  magnetized  molecules,  the  magnetic 
axes  of  the  molecules  being  to  a  greater  or  less  extent 
parallel.  If  this  were  so  the  north  pole  of  one  molecule 
would  be  counteracted  in  its  magnetic  effects  by  the  south 
pole  of  the  next,  and  the  only  molecules  capable  of  exert- 
ing external  magnetic  effects  would  be  those  on  the  sur- 
face, and  the  effect  is  exactly  the  same  as  would  be  pro- 
duced by  a  distribution  of  a  magnetic  matter  or  fluid,  or 
avoiding  the  idea  of  a  fluid,  a  distribution  of  magnetism 
over  the  surface  of  the  magnet.  The  amount  of  magnetism 
per  unit  area  is  called  the  magnetic  density.  If  the  magnet- 
ism is  regarded  as  being  uniformly  distributed  throughout 
the  mass  of  the  magnet,  the  quotient  of  the  magnetic  cur- 
rent by  the  volume  is  called  the  intensity  of  magnetization. 

Let  p  be  the  magnetic  density,  a  the  cross  section  and 
/  the  length  of  the  magnet.  Then  the  strength  of  pole 
is  m  —  p  a 

m  I  —  p  a  I 
ml 

•'• »  =  Tt 

_  Magnetic  moment 

Volume 

The  intensity  of  magnetization  and  magnetic  density 
are  therefore  practically  the  same,  the  one  presupposing  a 


32  NOTES   ON 


uniform  distribution  of  magnetism  throughout  the  mass, 
the  other  a  surface  distribution. 

If  the  magnetism  is  due  to  the  bar  being  situated  in  a 
magnetic  field,  the  intensity  of  magnetization  is  equal  to 
k  H,  k  being  what  is  called  a  "  coefficient  of  magnetiza- 
tion." A  few  values  of  k  are  given  in  §  340  ("  Thomp- 
son ").  Assuming  Barlow's  value  for  iron  32.8,  the  formula, 
intensity  of  magnetization  —  k  H  indicates  that  the  in- 
tensity of  magnetization  is  dependent  only  on  the  intensity 
of  the  field  ;  but  there  is  found  to  be  a  limiting  value  of 
magnetization  which  cannot  be  exceeded,  however  power- 
ful the  field  is.  This  is  stated  to  be  for  iron  1390  (p.  269, 
"  Thompson"),  and  the  strongest  field  that  could  be  util- 
ized in  magnetizing  iron  is  therefore  ^9°  =  42.4.  The 

value  of  this  coefficient  is,  however,  uncertain,  and  appears 
to  be  much  less  at  a  high  intensity  of  magnetization  than 
at  a  low 

30.  Solenoidal  Magnets. 

A  filament  of  magnetic  matter  so  magnetized  that  its 
strength  is  the  same  at  every  cross  section  is  called  a 
magnetic  solenoid.  A  long  thin  bar  magnet  uniformly 
magnetized  is  called  a  solenoidal  magnet,  or  simply  a 
solenoid,  in  distinction  to  a  magnetic  shell.  The  name 
solenoid  is  also  applied  to  a  helix  through  which  a  current 
passes.  (See  Note  42.)  As  the  magnet  poles  are  points  at 
which  the  magnetism  of  the  magnet  may  be  supposed  to 
be  concentrated,  and  from  which  magnetic  forces  act,  the 
potential  of  any  point  near  the  magnet  is  determined  by  its 

distance  from  the  two  poles,  or  V—  2 —  =  m  ( —,  j. 

The  exact  position  of  the  poles  is  difficult  to  determine, 
but  is  stated  in  g  122  to  be  in  long  thin  steel  magnets 


ELECTRICITY   AND   MAGNETISM. 


33 


about  rV  of  the  distance  from  the  end.  If  the  poles  are 
bent  to  meet,  forming  a  ring,  r  =  r'  for  all  external  points, 
and  there  is  therefore  no  potential  due  to  a  magnetized 
ring. 

31.  Potential  due  to  a  Magnetic  Shell. 

As  defined  in  \  107  ("  Thompson")  a  magnetic  shell  is  a 
thin  sheet  so  magnetized  that  the  two  sides  of  the  sheet 
have  opposite  kinds  of  magnetism.  The  demonstration 
and  use  of  the  expression  for  the  potential  due  to  a  mag- 
netic shell  requires  a  preliminary  definition  of  a  solid 
angle,  and  of  the  method  of  measuring  it.  The  solid 
angle  subtended  at  any  point  by  a  closed  curve  or  surface 
is  measured  by  the  area  of  a  sphere  of  unit  radius  de- 
scribed from  the  point  as  a  centre,  intercepted  by  lines 
drawn  from  all  parts  of  the  curve  to  the  point.  (See 
Fig.  64,  "  Thompson.")  As  the  areas  of  similar  surfaces 
on  spheres  are  as  the  square  of  the  radii  the  solid  angle, 
&>  =  area  on  unit  sphere,  A 

_  area  on  sphere  of  radius  r 

_  area  on  sphere  of  radius  r^ 
~^\~ 


To  compute  the  solid  angle. 
When  the  closed  curve  is  circular  and 
the  point  is  in  its  axis  it  is  necessary  only 
to  compute  the  area  of  a  zone  of  one  base 
on  a  sphere  of  unit  radius.  The  area  of 
the  zone  formed  by  the  revolution  of  AD 
around  AO  as  an  axis  is  (see  Chauvenet's 
Geometry,  Book  IX.,  Prop.  X.  Cor.  Ill) 

Ad  x  2  it  OA. 
But  Ad  =  A  O  —  dO  —  r  —  r  cos 

.'.  Area  =  zitr  (r  —  r  cos  6)  ; 
3 


Fig.  7. 


34 


NOTES   ON 


but  if  r  —  i,  area  is  solid  angle, 
.'.   GO  =  27t  (i  —  cos  6). 

To  calculate  the  potential. 

Let  r,  and  r2  be  the  distances  from  the  point  Z>  to  the  faces  of 
the  small  element  ds,  ft  be  the  angle  between  ds  and  its  projection 

,D 


Fig.  8. 

perpendicular  to  r1  and  p  be  the  magnetic  density.    The  strength 
of  the  shell  i  being  the  product  of  the  density  and  thickness, 

-     i  =  pt   .-.  p  =  -. 


The  quantity  of  magnetism  on  the  small  element  ds  is 


dm  =  ds.p  =  ds. — 


(I) 


The  potential  at  D  due  to  ds  is 

-       *=<*-„).    .    .    (a) 


ELECTRICITY   AND   MAGNETISM.  35 


But  cos  ft  =  — — ; — -1.     Substituting  this  value  and   that  of  dm 


rfF=™-cos/S (3) 

r~ 

But  as  ds  is  an  infinitely  small  element,  its  plane  projection 
ds  cos  ft  perpendicular  to  rl  is  sensibly  equal  to  the  area  on  a 
sphere  of  radius  r.  The  solid  angle,  therefore,  subtended  at  D 

by  the  element  ds  is 

_  ds.  cos  ft 
f" 

Substituting  in  (3) 

dV  =  deal 

V  —  ooi. 


32.    Equipotential    Surfaces    and    Lines    of   Force    of 
a  Magnetic  Shell. 

The  computation  of  the  solid  angle  is,  as  shown,  simple 
when  the  point  D  is  on  the  axis  normal  to  the  shell  at  its 
centre,  but  when  D  is  oblique  the  area  becomes  an  ellipse 
on  a  spherical  surface  of  unit  radius,  and  the  calculation 
is  extremely  difficult.  From  the  formula,  however,  for 
magnetic  potential  a  few  relations  are  readily  deduced. 
As  at  all  points  where  the  shell  subtends  the  same  solid 
angle  the  potential  is  the  same,  any  equipotential  sur- 
face is  evidently  most  remote  from  the  shell  on  the  axis 
normal  to  its  centre.  As  the  point  of  view  becomes 
oblique  it  must  approach  the  shell  that  the  solid  angle 
may  be  the  same,  and  at  all  points  in  the  plane  of  the  shell 
the  solid  angle,  and  consequently  the  potential,  are  zero. 
The  general  form  of  the  equipotential  surfaces  is,  there- 
fore, that  of  deep  bowls  concave  to  the  shell,  and  most 
remote  from  it  on  its  perpendicular  axis.  At  a  point  close 
to  the  shell  the  solid  angle  is  a  hemisphere  or  2Tt  and  the 


36  NOTES  ON 


potential  27rz".  On  the  opposite  side  the  potential  is  —  27T/, 
or  47T*  ergs  must  be  expended  in  moving  unit  pole  from 
one  side  of  the  shell  to  the  other.  If  the  equipotential 
surfaces  indicate  unit  difference  of  potential  there  are, 
therefore,  ^Tti  surfaces.  From  the  equipotential  surfaces 
the  direction  of  the  lines  of  force  may  be  traced,  as  they 
start  from  the  side  of  the  shell  having  north  polarity  and 
curve  so  as  to  cut  each  surface  at  right  angles,  finally  en- 
tering the  south  pole  of  the  shell  at  right  angles. 

33.  Work  Done  in  Moving  Pole  near  Shell. 

Potential  being  measured  by  the  work  done  on  unit 
pole  in  bringing  it  up  to  a  point  from  an  infinite  distance, 
the  work  done  on  a  pole  of  strength  in  is  mcoi.  It  is  pos- 
sible under  the  conventions  made  as  to  the  number  of 
lines  of  force  to  express  this  in  another  way.  As  already 
shown,  the  number  of  lines  of  force  given  off  by  a  pole  of 
strength  ;;/  is  ^nm,  but  as  these  radiate  in  all  directions, 
they  are  given  off  throughout  a  solid  angle  47?  subtended 
at  the  centre  of  a  sphere.  Through  any  solid  angle  &?  the 
number  of  lines  is,  therefore,  moa.  Calling  this  number  N, 
the  above  expression  becomes  Ni,  or  the  work  done  in 
bringing  a  pole  up  to  a  position  near  a  magnet  shell  is 
measured  by  the  product  of  the  strength  of  the  shell  and 
the  number  of  lines  of  force  of  the  pole  cut  by  the  shell. 
This  is  evidently  a  measure  of  the  work  done  either  in 
bringing  the  pole  up  to  the  shell  or  the  shell  to  the  pole, 
and  is,  therefore,  sometimes  called  the  mutual  potential 
of  the  pole  and  shell.  The  work  done  in  bringing  the 
pole  from  infinity  to  a  point  where  it  intercepts  Nl  lines  is 
Nil.  If  now  it  be  moved  to  another  in  which  it  intercepts 
N*  lines,  the  work  done  between  the  points  is 

Work  done  =  i  (N.,  -  ,V,). 


ELECTRICITY   AND   MAGNETISM.  37 

The  difference  of  potential  between  the  points  is 

i  (Ni  —  AT,) 
—  ^       —  -' 


or 


magnetic  potential  being  the  quotient  of  work  done  in 
moving  pole,  by  the  strength  of  pole. 

The  work  done  may  be  either  positive  or  negative  and 
the  above  expression  may,  therefore,  have  either  sign.  If  A7! 
>  A2  the  work  done  in  passing  from  N\  to  A2  is  negative, 
or  the  shell  tends  to  move  in  such  a  direction  as  to  include 
a  minimum  number  of  lines  of  force.  As  these  pass  in  the 
positive  direction,  exactly  the  same  relation  is  expressed 
by  saying  that  a  magnetic  shell  in  a  field  tends  to  place 
itself  so  as  to  enclose  the  maximum  number  of  negative 
lines  of  force.  If  the  north  pole  of  a  magnet  shell  is 
brought  up  to  the  north  pole  of  a  magnet,  this  relation  is 
readily  seen,  as  the  shell  will  be  repelled  into  a  position 
in  which  it  will  enclose  as  few  lines  of  force  taken  in  the 
positive  direction  as  is  possible.  If  the  same  face  be  ap- 
proached to  a  south  pole,  it  is  attracted  and  moves  into  a 
position  in  which  the  maximum  number  of  lines  cut  the 
shell  in  the  negative  direction. 

34.  Equivalent  Magnetic  Shells. 

The  relations  deduced  for  magnetic  shells  are  of  great 
service,  as  they  are  applicable  to  the  case  of  a  voltaic 
circuit  in  a  magnetic  field.  If  a  wire  carrying  a  current 
be  looped  into  a  circle,  the  lines  of  force  which  ordinarily 
encircle  the  conductor  combine  to  act  in  the  same  direc- 
tion on  a  pole  at  a  distance  from  the  circuit.  Thus  in 
Fig.  86  ("  Thompson  "),  it  is  seen  that  all  the  lines  of  force 
due  to  the  current  pass  in  the  same  direction  through  the 
plane  of  the  circuit  as  do  those  of  a  magnetic  shell.  A 
closed  voltaic  circuit  in  a  magnetic  field,  as  may  be  readily 


38  NOTES  ON 


shown  by  experiment,  is  acted  upon  as  a  magnet  would 
be.  It  is  found  that  the  magnetic  effects  of  the  north  pole 
of  a  magnet  are  identical  in  nature  with  those  of  a  circuit, 
in  which  the  current  flows  in  a  direction  opposite  to  that  in 
which  the  hands  of  a  watch  move.  This  direction  is 
known  as  the  negative  direction  of  the  current,  and  the 
magnetic  effects  of  a  positive  pole  and  of  a  negative  cur- 
rent are,  therefore,  similar.  Looking  at  the  other  side  of 
the  loop,  the  current  would  appear  to  pass  in  the  direction 
in  which  the  hands  of  a  watch  move,  or  in  the  positive 
direction  ;  but  if  a  north  pole  be  approached  to  the  circuit 
from  the  side  on  which  the  current  appears  to  have  this 
direction  it  is  attracted,  showing  that  a  positive  current 
produces  magnetic  effects  similar  to  those  of  a  negative 
pole.  As  the  direction  in  which  a  north  pole  moves  shows 
the  direction  of  the  lines  of  force,  it  is  seen  from  the  above 
that  the  lines  of  force  enter  that  face  of  the  plane  of  the 
circuit  in  which  the  current  appears  to  move  "  with  the 
sun,"  or  in  the  positive  direction,  and  emerge  from  the 
other  face.  As  it  is  a  matter  of  great  importance  to  be 
able  to  connect  the  direction  of  the  lines  of  force  with  that 
of  the  current  to  which  they  are  due,  several  rules  have 
been  given,  one  of  the  best  of  which  is  the  comparison 
between  the  direction  of  rotation  of  a  corkscrew  and  that 
of  the  motion  of  its  point.  If  the  wrist  be  rotated  in 
the  right-handed  direction,  the  point  advances ;  and  con- 
sidering the  motion  of  the  wrist  to  be  that  of  the  current, 
the  movement  of  the  point  corresponds  to  that  of  a  north 
pole,  and  indicates  the  direction  of  the  lines  of  force. 
This  relation  is  said  to  be  that  of  "  right-handed  cyclical 
order,"  and  the  direction  of  the  current  and  of  the  lines  of 
force  are  spoken  of  as  being  thus  related. 

The   magnetic  action  of  a  voltaic  circuit   is   found  to 
depend  upon  the  strength  of  the  current,  and  on  the  area 


ELECTRICITY  AND   MAGNETISM.  39 


of  the  enclosed  surface.  It  is,  therefore,  evident  that  for 
every  closed  circuit,  a  magnetic  shell  whose  edges  coin- 
cided in  position  with  the  circuit  could  be  substituted,  if  a 
certain  relation  were  established  between  the  units  measur- 
ing the  strength  of  the  current  and  the  strength  of  the 
shell.  This  relation  is  that  expressed  in  the  definition  in 
g  195.  The  absolute  electromagnetic  unit  of  current  is 
that  current  which  in  passing  through  a  conductor  one 
centimetre  long,  bent  so  as  to  be  in  all  parts  distant  one 
centimetre  from  a  unit  pole,  acts  on  the  pole  with  a  force 
of  one  dyne.  By  this  definition  unit  current  produces 
unit  field  at  unit  distance.  But  so  does  a  shell  or  pole  of 
unit  strength.  By  expressing,  therefore,  the  current  in 
these  units,  the  magnetic  effect  of  the  current  is  the  same 
as  that  due  to  a  magnetic  shell  whose  edges  coincide  with 
the  circuit,  and  whose  strength  is  equal  to  that  of  the  cur- 
rent. This  is  called  an  equivalent  magnetic  shell,  and 
all  relations  hitherto  traced  for  the  shell  are  now  applica- 
ble to  the  closed  circuit. 

35.  Potential  due  to  a  Closed  Voltaic  Circuit. 

The  potential  due  to  a  current  at  a  point  is  therefore 
Coo,  where  Cis  the  current  measured  in  absolute  units, 
and  GO  is  the  solid  angle  subtended  by  the  circuit  at 
the  point.  As,  however,  a  positive  current  produces  the 
same  magnetic  effects  as  a  negative  pole,  the  sign  of  the 
potential  is  always  the  opposite  of  that  of  the  current,  or 


The  difference  of  potential  between  two  points  is 


The  magnetic  force  due  to  a  current  at  a  distance  x  is 
(Note  14) 


40  NOTES  ON 


dx 

The  potential  is  iitC  on  one  side  of  the  circuit  and 
—  2.TfC  on  the  other,  changing  by  4?rC.  Hence  there 
are  4;rCequipotential  surfaces. 

36.  Work  Done  in  Moving  a  Circuit  Near  a  Pole. 

This  is  a  problem  of  the  greatest  importance,  as  it  under- 
lies the  action  of  the  dynamo  machine.  As  already 
traced  (Note  33),  the  work  done  in  moving  a  magnetic 
shell  near  a  pole,  or  conversely  the  pole  near  the  shell,  is 

Work  done  =  moot  =  Ni. 

Similarly  the  work  done  in  moving  a  closed  circuit  near 
a  pole  is 

Work  dons  =  —  moaC  =  —  NC, 

N  being  the  number  of  lines  of  force  of  the  pole  passing 
through  the  circuit.  If  the  circuit  be  brought  up  from  an 
infinite  distance  to  a  point  where  it  intersects  /V,  lines  due 
to  the  pole  the  work  is  —  JV}  C.  If  now  moved  still  farther 
so  that  it  intersects  a  greater  number,  /V2,  the  work  done 
between  the  points  is 

V/ork  = 
and 

Difference  of  potential  =  -  C(A/2  ~  N$  =  -  C(oo,  -  GO,). 

If  TVa  >  N\  the  work  is  negative,  and  the  circuit  tends 
to  move  therefore  in  such  a  manner  as  to  make  the  num- 
ber of  lines  enclosed  a  maximum.  If  a  circuit  be  placed 
in  a  magnetic  field  so  that  the  lines  of  the  field  while 
parallel  to  those  of  the  current  pass  in  the  opposite  direc- 


ELECTRICITY  AND   MAGNETISM.  41 

tion,  the  circuit  will,  if  free,  first  turn  to  bring  the  lines  in 
the  same  direction,  and  will  then  move  to  make  the  num- 
ber enclosed  a  maximum  (Fig.  87,  "Thompson"). 

It  may  be  useful  to  obtain  an  expression  for  the  work 
done  in  this  last  case,  as  an  understanding  of  the  theory 
will  assist  in  the  comprehension  of  the  working  of  electric- 
motors.  Imagine  a  closed  circuit  placed  in  a  uniform 
field.  If  the  circuit  be  moved  parallel  to  itself,  the  num- 
ber of  lines  enclosed  is  constant,  and  consequently  no  work 
is  done  in  whatever  direction  the  circuit  be  moved.  If, 
however,  the  coil  be  rotated  on  an  axis  in  its  plane,  it  will 
enclose  a  varying  number.  If  0  be  the  angle  between  the 
normal  to  the  plane  of  the  coil  and  the  direction  of  the 
lines  of  force  of  the  field,  and  the  number  of  lines  passing 
through  the  coil  when  its  plane  is  perpendicular  to  them 
be  Nt  the  number  enclosed  when  at  an  angle  6  is  N  cos  0. 
If  the  angle  be  now  changed  to  6'  the  number  enclosed  is 
JV  cos  V ,  and  the  work  done  in  passing  from  one  position 
to  the  other  is 

Work  =  -  C(Ncos  0'  -  NCOS  6). 

Suppose  that  the  coil  be  rotated.  The  work  done  is 
easily  calculated  : 

In  the  first  quadrant, 

work  =  -  C  (Ncos  90°  -  ^Ycoso0)  =  CN\ 
in  the  second  quadrant, 

work  =  -  C(A^cos  180°  -  N  cos  90°)  =  CN; 
in  the  third  quadrant, 

work  =  —  C(A^cos  270°  —  A^cos  180°)  =  —  CN\ 
and  in  the  fourth  quadrant, 

work  =  -  C(/Vcoso°  -  A7  cos  270°)  =  -  CN. 

In  the  first  half  of  the  revolution,  therefore,  work  equal 
to  2  CN  has  to  be  done  in  order  to  more  the  coil,  but  in 


42  NOTES   ON 


the  latter  half  the  coil  will  do  the  same  amount  of  work. 
The  potential  energy  of  the  coil  is,  therefore,  greatest  when 
0  =  1 80°,  or  when  the  lines  of  force  of  the  field  are  parallel 
to  those  of  the  coil,  but  in  the  opposite  direction,  and  if 
the  coil  be  then  left  free  to  move,  it  will  rotate  to  make 
6  —  0°,  doing  work  equal  to  2QV,  and  then  requiring 
work  to  be  done  on  it  to  cause  further  movement.  On 
the  supposition  already  stated  that  the  field  is  uniform, 
N  =  HA,  H  being  the  intensity  of  the  field  and  A  the 
area  of  the  coil.  The  work  done  by  the  coil  in  rotating 
through  two  quadrants  may  then  be  expressed  as  2CHA, 
this  also  measuring  the  work  done  on  the  coil  in  the  other 
two  quadrants. 

As  a  resume  of  the  above,  we  have  the  rule,  that  a  mov- 
able circuit  in  a  magnetic  field  tends  to  place  itself  so 
as  to  enclose  the  maximum  number  of  lines  of  force  in 
right-handed  cyclical  order. 

37.   To  Calculate  the  Intensity  of  the  Field  due  to  a 

Voltaic  Circuit. 

The  force  acting  on  unit  pole,  or  the  intensity  of  the  field,  is 
by  Note  35  the  rate  of  change  of  po- 
tential per  unit  of  length.  The  in- 
tensity of  field  at  a  distance  x  is 


The  difficulty  of  calculating  the 
value  of  oo  makes  the  general  solu- 
tion extremely  complicated.  It  is, 
however,  easy  to  calculate  the  inten- 
sity of  the  field  at  any  point  on  the 
P.  axis  of  the  circuit,  as  in  that  case 

GO  —  2it  (i  —  cos  6). 
Let  x  =  the  distance  of  the  point  A  from  the  circuit. 
r  r=  radius  of  the  coil. 


ELECTRICITY   AND   MAGNETISM. 


43 


Then  dV  =  —  zrtC.d  (L  —  cos  0)       cos  0  = 


dx 


2.x Cr* 


At  the  centre  of  the  circle  the  force  is  a  maximum,  and  is 
•  = as  in  §  195.     The  —  sign  shows  that  with  a  posi- 

tive current  the  force  is  one  of  attraction. 


38.  Position  of  Equilibrium  of  a  Circuit  and  Magnet. 

Consider  the  magnet  as  composed  of  two  poles  of  strength 
m   and    —  m    connected    rigidly.         -. 
The  formula  for  the  force  in  the 
field  may  be  written 


= sin3  0. 


The   forces   acting  on   the  two 
poles  are 


and 


/I 

1 

1 

\                    ^v. 

e  X             &  >» 

/ 

A                      B 

---  sin 

r 


Fig.  10. 


The  resultant  force  is 


~  (sin^  0  -  sin'  G'). 

This  is  zero  when  0  =  0',  or  when  the  centre  of  the  magnet 
is  at  the  centre  of  the  coil.  In  any  other  position  there  will  be 
a  force  acting  and  the  equilibrium  will  be  unstable.  If,  there- 
fore, either  the  coil  or  magnet  is  free  to  move,  the  coil  will 


NOTES  ON 


place  itself  so  that  the  middle  of  the  magnet  is  at  its  centre. 
(See  Fig.  87,  "  Thompson.") 

39.  Mutual  Potential  of  two  Circuits  (§  320). 

The  work  done  in  bringing  one  circuit  up  to  another,  or 
the  "  mutual  potential  "  of  the  two  circuits  is,  as  given  in 

,    cos  s 
|  320,  —  cc  .  ss  .      This  expression  is  one  of  great 

theoretical  importance,  but  its  derivation  is  difficult  and 
out  of  place  in  an  elementary  treatise. 

The  work  done  in  moving  a  circuit  near  a  pole  or  in  a 
field  has  already  been  shown  to  be  —  NC,  and  it  is 
obviously  immaterial  whether  the  lines  offeree  TV  are  due 
to  a  pole,  a  magnetic  shell  or  another  circuit.  Suppose 
two  circuits  A  and  B,  carrying  currents  of  strength  C 
and  C,'  and  let  N^  be  the  number  of  lines  of  force 
due  to  A  enclosed  by  B,  and  N*  the  number  due  to  B 
enclosed  by  A.  If  B  is  moved  out  of  the  field  caused 
by  A  the  work  done  is  A^  C  If  A  is  now  moved  so  as  to 
resume  its  former  relative  position  to  B  the  work  done  is 
—  N^  C'.  The  coils  are  now  in  the  same  relative  position 
as  at  first  and  if  there  are  no  external  magnetic  forces,  no 
work  can  have  been  done  in  moving  the  system.  Hence 

Ni  C-Nn  C  =  O. 


If  the  current  in  each  is  of  unit  strength 

.M  =  Nv 

or  each  encloses  the  same  number  of  the  other's  lines  of 
force.     Returning  to  the  expression  for  the  work  done. 

CC  ^-£  ss',  and  making  Cand  C'  each  unity,  the  number 
of  lines  enclosed  by  each  is  —  -  ss',  and  this  number  may 


ELECTRICITY   AND   MAGNETISM.  45 

be  represented  by  the  single  symbol  J/and  is  dependent 
only  on  the  position  and  areas  of  the  two  coils. 

Let  the  planes  of  the  two  circuits  be  parallel,  and  the 
current  flowing  in  the  same  direction  in  each,  that  in 
which  the  hands  of  a  watch  move.  The  negative  sign  of 
the  formula  shows  that  the  circuits  attract  each  other,  and 
this  is  also  evident  from  Maxwell's  rule  that  a  voltaic  cir- 
cuit free  to  move  always  places  itself  so  as  to  enclose  the 
greatest  possible  number  of  lines  of  force.  The  nearer  the 
circuits,  the  greater  the  value  of  M,  and  if  they  become 

coincident  M  would  be  — ,  or  infinite.   As,  however,  r  will 
o 

always  have  a  finite  value,  the  maximum  value  of  M 
exists  when  the  coils  are  touching,  or  r  is  a  minimum. 
As  the  coils  tend  to  approach  or  to  diminish,  r,  the 
"  coefficient  of  mutual  potential  "  M,  always  tends  to  a 
maximum.  This  quantity  is  hereafter  referred  to  (Note 
63)  as  the  "  coefficient  of  mutual  induction." 

40.  Conversion  of  Units  (§  324). 

The  use  of  the  dimensions  of  units  in  passing  from  one 
system  to  another  has  been  illustrated  in  Note  19.  In 
electrical  calculations,  the  most  frequent  change  to  be 
made  is  that  from  the  C.G.S.  system  to  the  British  units 
based  on  the  foot,  grain,  and  second.  The  ratios  between 
these  units  are  shown  in  the  following  table  from  Jenkin's 
"  Electricity." 


46 


NOTES   ON 


i 

a 

Number          £ 
of  C.G.S.        5   • 
Units  in          j^J 
one  British        bco 
Unit  (^4).        ^ 

f 

Number 
of  British 
Units  in 
one  C.G.S. 

Unit  (B). 

g     Mass 

o  BTT,-£.»S 

|  r  Length  . 

. 

"O 

Time..  . 

r 

1.1' 

|]  Force  ._ 

p 

— 

1; 

["Work 

* 

60.198 

1.7795820 

2.2204179 

0.01661185 

;*' 

Quantity  ... 

? 

42.8346 

1.631794912.3682051 

0.0233456 

3 

Current. 

*  or 
C 

42.8346 

1.6317949 

2.3682051 

0.0233456 

^ 

g 

Potential  

V 

1.40536       0.14778741.8522125 

o.7Ti=;6i 

i 

Resistance  

0.03280899  2.5159929  i.  4040071^0.4704!; 

w 

Capacity  

r 

30.47945          !l-4S4007i;2.^I!?QQ2Q 

0.03280899 

•r 

Strength  of  Pole... 

m 

42.8346 

1.6317949 

2.3682051 

0.0233456 

gl  Magnetic  Potential. 

V 

1.40536 

0.1477874  1.8522125 

0.711561 

g     Intensity  of  Field... 

H 

0.0461085 

2.6637804 

1.3362196 

21.6880 

* 

o 
c 

Current  

i  or 
C 

1.40536 

0.1477874 

1.8522125 

0.711561 

i 

Quantity  ...  j  @ 

1.40536 

0.1477874 

I.8522I25 

0.711561 

0 

-Potential  » 
Electromotive  F'ce  f 

F 
E 

42.8346 

1.6317949 

2.3682051 

0-0,33456 

o 

Resistance.  .  . 

E> 

Q 

Q    o 

5 

Capacity  

c 

30.47945        IJ^J^^/i 
0.0328089912.  5159929 

1.4840071 

30.47945 

The  following  table,  showing  the  relations  between  the 
practical  units  in  common  use,  may  be  convenient  for 
reference : 

i  metre  =  39.37043  inches  =  3.28087  feet. 
I  kilogramme  =  2.20462  avoirdupois  pounds. 
i  kilogrammetre,  or  i    kilogramme    raised   one   metre 
per  second  =  7.23307  foot-pounds  per  second. 


ELECTRICITY  AND    MAGNETISM.  47 

i   Force  cle  cheval   or   French    H.   P.  =  75  kilogram- 
metres  per  second. 

I  English  H.  P.  =  33000  foot-pounds  per  minute. 
=  550         "         "         "    second. 
:=  76.04  kilogrammetres  per  second. 
=  1.014  force  de  cheval, 
I  gramme  —  981  dynes  (980.868  at  Paris), 
i  gramme  centimetre,   or  I   gramme  raised  one  centi- 
metre in  a  second  =  981  ergs. 

I  pound  avoirdupois  =  4.45  x  io5  dynes  nearly. 
I  foot-pound  per  second  =  1.356  x  io7  ergs  nearly. 
i  Volt  =  io8  absolute  electromagnetic  units  of  potential, 
i  Ohm  —  io9       "      electromagnetic  units  of  resistance. 
Practical  Ohrn  =  .9895  x  io9  absolute  units  (Lord  Ray- 
leigh). 

I  Ampere  =  nj  of  an  absolute  electromagnetic  unit  of 
current. 

i  Coulomb  =  iV  of  an  absolute  electromagnetic  unit  of 
quantity. 

=  I  ampere  per  second. 

i  Farad  =  io~9  absolute  electromagnetic  units  of  ca- 
pacity. 

i  Watt  (see  Note  55). 

=  IOT  ergs  =  .7373  foot-pounds, 
=  TJB-  English  H.  P. 

I  thermal  unit  =  i  gramme  of  water  raised  i°  C. 
I  Joule  =  io7  ergs  =  .24  thermal  unit  (See  Note  55). 
Mechanical  equivalent  of  heat  =  772  foot-pounds  i°  F. 

=  1390-      ••      rc. 

Same  in  metric  system  =  424  kilogrammetres  i°  C. 

=  42400  gramme-centimetres  i°  C. 

=  4.16  x   io7  ergs. 
i  Siemens  unit  =  .9536  practical  ohm. 


48  NOTES   ON 


i  Jacobi  =  current   evolving  i    Cm '.  of  mixed  gas   per 
minute  at  o°  and  760  mm. 
=  .095  amperes. 

At  any  place  the  weight  of  the  gramme  is  equal  to  g 
dynes.  The  value  of  g  for  any  latitude  may  be  found 
approximately  from  the  formula 

g  —  980.6056  —  2.5028  cos  2/1  —  .000003^, 

A  being  the  latitude  and  h  the  height  above  sea  level.  The 
limiting  values  of  g  are  978.1  at  the  equator  and  983.1  at 
the  poles. 

41.  Determination  of  the  Horizontal  Component  of  the 
Earth's  Magnetism  (§  325  a). 

The  time  of  vibration  of  a  particle  acted  on  by  a  con- 
stant force  is  /  =  it  — ,  /  being  the  time  of  a  half  or  sim- 
ple vibration,  and  ju  the  acceleration.  The  latter  is  in  any 
case  equal  to  the  moment  of  the  impressed  forces  divided 
by  the  moment  of  inertia.  When  the  arcs  of  vibration 
are  small  this  may  be  applied  to  a  magnet  oscillating  in  a 
uniform  field  and  the  time  of  a  complete  or  double  oscilla- 
tion of  a  magnet  is  therefore 


K  being  the  moment  of  inertia.  M  the  magnetic  mo- 
ment and  //the  horizontal  intensity. 

A, — To  make  the  observation,  a  magnet  is  allowed  to 
oscillate  and  the  time  of  vibration  is  determined  as  ac- 


ELECTRICITY  AND   MAGNETISM.  49 

curately  as  possible.  This  is  best  done  by  determining  the 
approximate  time  of  one  oscillation,  and  allowing  the 
magnet  to  oscillate  a  known  time.  Dividing  this  time  by 
the  approximate  time  of  one  oscillation  gives  the  approxi- 
mate number  of  oscillations.  Taking  the  nearest  whole 
number  to  this,  and  dividing  the  whole  time  by  it  gives 
the  exact  time  of  one  vibration.  If  the  approximate  num- 
ber of  vibrations  fell  midway  between  two  whole  numbers, 
the  observation  would  have  to  be  repeated  until  it  was 
known  with  certainty  how  many  oscillations  had  been  made 
in  the  observed  time.  The  oscillations  must  be  of  small 
amplitude,  and  in  very  exact  observations  must  be  reduced 
to  an  infinitely  small  arc.  If  possible,  the  magnet  should 
be  supported  by  a  single  fibre  to  avoid  torsion,  but  if  a 
wire  has  to  be  used,  the  torsion  must  be  allowed  for.  If 
the  magnet  is  of  simple  form,  either  bar  or  cylindrical,  the 
value  of  K  may  be  determined  from  the  formulas  ing  325^, 
but  if  these  do  not  apply,  A'  may  be  determined  by  observa- 
tion of  the  time  of  vibration  of  the  magnet,  and  of  the 
time,  /,,  when  its  moment  of  inertia  is  increased  by  the 
addition  of  a  weiht  of  known  moment  Kl. 


From  (i)  the  value  ofMffis  -  .     .     .     (2) 

•i 

When  the  weight  is  added,  MH  =  ^       ,+  ^ 


or  K  = 


Knowing  A"  it  is  possible  to  compute  Mfffrom  (i). 
In  the  case  of  the  oscillating  magnet,  the  magnet  and 
earth  acted  mutually  on  each  other.     In  order  to  obtain 

the  ratio  T-^the  force  of  the  magnet  must  act  against  that 


NOTES  ON 


of  the  earth,  and  this  is  done  by  measuring  the  deflection 
from  the  magnetic  meridian  that 
the  magnet  will  cause  in  a  small 
magnetic  needle  near  it.  A'S  is 
the  magnet  the  time  of  whose 
oscillation  has  been  determined, 
and  it  is  placed  at  right  angles  to 
the  magnetic  meridian,  so  that 
its  centre  is  due  north  or  south  of 
the  small  needle  at  O.  Let  r  be 
the  distance  of  either  A^or  S  from 
(9,  and  m  -be  the  strength  of  N 

N1  ^S  and  S.  Then  the  force  exerted 

by  S  on  a  south  pole  in1  at  O  is 


Fig.  ii. 


—  in  the  direction  Oa,  which  may  be  taken  to  represent 

it    in   magnitude    and   direction.      Similarly    Oc   would 
represent  the  force  of  attraction  of  the  pole  N.     The  re- 
sultant force  is  Ob: 
From  similar  triangles 

Oa  :  Ob  :  :  So  :  NS ; 
or  calling  Ob,  T,  and  NSt  L, 

mm'                                         mm'  L 
•—t-:T::r:L      .\T- -=—. 


Let  M'  =  ;«'/',  the  moment  of  the  small  magnet,  and 
M  =  mL  the  moment  of  the  large  one.  Then  the 
couples  acting  on  the  small  needle  when  it  has  a  perma- 
nent deflection  0  are  ; 

,  a     ±  <***  mm  LI'  MM' 

to  deflect,  77  cos  0  = cos  6  =•  — -3 —  cos  6  ; 

to  retain  in  the  meridian,  m'l'H sin  0  =  M' //sin  0. 
Equating  these  moments  and  reducing 


ELECTRICITY   AND   MAGNETISM.  51 

^  =  r3tan0  .......    (3) 

By  combining  (2)  and  (3) 


J3,  —  In  the  Kew  Magnetometer  the  deflecting  magnet  is 
placed  east  or  west  of  the  small  magnet  instead  of  north 
or  south,  and  the  formula  is  different. 

Let  r  be  the  distance  between  the  pole  s  and  the  centre 

;«* 


Fig.  12. 


of  the  deflecting  magnet,  L  be  the  length  of  the  deflecting 
magnet,  m  the  strength  of  pole  of  the  deflecting  and  m'  that 
of  the  deflected  magnet,  and  /'  the  length  of  the  latter.  The 

mm' 
force  of  repulsion  exerted  by  5  on  j  is  --  —  -  ,  r  be- 


ing  great  in  comparison  with  L.     The  force  of  attraction 
of  TV  on  s  is  —    -  —  .     These  forces  may  be  considered 


as  acting  in  the  same  line  but  in  opposite  directions,  and 
the  resultant  force  acting  on  s  is 


=  mm' 


z 
r  — 


_L  _  \ 
L\*  } 

r  +  -  )  I 

a/ 


52  NOTES  ON 


(5) 


The  moment    of   the  deflecting   couple   on   the  small 
magnet  is 

FT  cos  0  =  2AfM'  . cos  0  , 

r-?)2 

or  since  as  above,  L  is  small  in  comparison  with  r,  its 
square  and  higher  powers  may  be  neglected,  and  the 
moment  of  the  couple  is 

*MM'  .- !l_cosfl  = 


4 

The  moment  of  the  couple  tending  to  retain  the  small 
magnet  in  the  meridian  is 

M'Hsm  0. 
Equating  and  reducing 

^=ir'tan0 (6) 

C. — This  derivation  contains  many  assumptions.  A 
more  rigorously  correct  formula  is  that  given  by  Kohl- 
rausch, 

M  —  i  r*  tan  9  ~~  ri*  tan  6' 

7f  ~  a  ~    ""f^^C 

In  which  r  and  ri  are  the  distances  between  the  centres 
of  the  magnets  in  two  successive  positions,  and  0  and 
0'  the  corresponding  angles  of  deflection.  The  deflectir.B 
magnet  is  placed  east  or  west  of  the  deflected  needle  as 
in  the  last  case. 


ELECTRICITY   AND    MAGNETISM.  53 

The  formula  of  (6)  is  the  one  generally  used  in  work  with 
the  Kew  Magnetometer,  but  is  true  only  when  r  is  large  in 
comparison  with  /,.  The  more  accurate  formula  can  be  readily 
derived.  From  equation  (5) : 


r' 
The  moment  of  the  deflecting  couple  is 


zMM' 
Fl   cos6  =  T-5T-  Li  +  -  '  ~?   )cos 


Equating  this  with  M'  H  sin  0,  the  moment  tending  to  retain 
the  couple  in  the  meridian,  and  reducing, 


By  repeating  the  observation  by  placing  the  deflecting  mag- 
net so  that  its  centre  is  at  a  distance  of  r\  from  the  needle,  a 
new  deflection,  0',  is  obtained,  and 


Multiplying  (7)  by  r5  and   (8)  by  r,8,  and  subtracting  the  lat 
ter  from  the  former, 

r*  tan  0  -  r,6  tan  C'  =  2  .  ^  (r1  -  r  2)  ; 
H 

or, 


54  NOTES  ON 


IV.  MEASUREMENTS  AND    FORMULAE. 

42.  Solenoids  (§  327).  Ampere's  Theory  of  Magnetism 
<§  338). 

As  stated  in  \  327,  a  spiral  coil  of  wire  through  which 
a  current  passes  is  called  a  solenoid.  The  definition  has 
already  been  given  as  that  of  a  magnetic  filament  uni- 
formly magnetized,  and  as  a  spiral  coil  carrying  a  current 
exerts  the  same  forces,  and  is  similarly  acted  upon  in  a 
magnetic  field,  it  is  called  by  the  same  name.  Theo- 
retically the  turns  of  the  coil  should  be  exactly  parallel, 
and  at  right  angles  to  the  longitudinal  axis,  but  as  this  is 
impossible  the  ends  of  the  helix  are  brought  in  through 
the  coil  from  each  end  to  the  centre  as  in  Fig.  116.  The 
current  flowing  in  these  branches  exerts  an  effect  equal 
and  opposite  to  that  due  to  the  longitudinal  component  of 
the  spiral,  and  the  resultant  effect  of  a  solenoid  thus  con- 
structed is  that  of  a  number  of  parallel  turns  only,  or  "* 
the  theoretical  solenoid. 

If  the  helix  be  free  to  move  it  will,  when  a  current  is 
passed  through  it,  move  so  as  to  include  the  maximum 
number  of  lines  of  force  in  the  field.  In  the  earth's  field,  it 
therefore  assumes  the  same  position  as  the  dipping  needle. 
The  end  pointing  north  is  called  the  north  pole  of  the  sole- 
noid, as  with  the  ordinary  bar  magnet.  Let  a  solenoid 
be  suspended  so  as  to  move  freely,  and  a  magnet  be 
brought  near  it  so  that  the  north  poles  are  nearest.  The 
solenoid  will  be  repelled,  but  if  after  repulsion,  the  south 
pole  of  the  magnet  is  brought  up,  the  north  pole  of  the 
solenoid  is  attracted.  Magnetic  forces  are  evidently  act- 
ing between  the  coil  through  which  a  current  is  flowing 


ELECTRICITY   AND    MAGNETISM. 


and  the  piece  of  steel  which  we  call  a  magnet.  Many 
other  similar  effects  have  been  alluded  to,  and  they 
suggested  to  Ampere  a  theory  of  magnetism,  which  ex- 
plains very  many  peculiar  relations  and  is  of  great  prac- 
tical utility.  He  conceived  that  a  magnet  was  composed 
of  a  great  number  of  molecules,  around  each  of  which 
flowed  an  electric  current  in  a  constant  direction.  In  an 
ordinary  unmagnetized  bar  of  steel,  these  currents  lie  in 
all  possible  planes,  so  that  their  resultant  magnetic  effect 
is  zero.  If,  however,  the  bar  be  magnetized,  the  energy 
expended  in  so  doing  operates  to  turn  the  molecules  so 
that  the  currents  are  now  parallel.  In  looking  at  the  end 
of  a  magnet,  the  molecu- 
lar currents  would,  as 
shown  in  the  figure,  coun- 
teract each  other  in  the 
substance  of  the  magnet, 
but  the  current  on  the 
outer  edge  of  the  outer 
row  of  molecules  being 
unbalanced  would  cause 
a  resultant  current  on  the 
surface  in  the  direction 
opposite  to  the  motion  of 
the  hands  of  a  clock.  In 
a  solenoid  a  current  flows  in  this  direction  when  looking 
at  the  north  pole  of  the  solenoid,  and  the  figure  therefore 
shows  the  theoretical  condition  of  the  north  pole  of  the 
magnet.  Looking  at  the  other  face  the  currents  appear  to 
flow  in  the  opposite  direction.  From  these  suppositions 
follows  the  rule  given  on  p.  284.  The  theory  explains  many 
peculiar  effects,  such  as  those  referred  to  in  \  112  and 
|  113,  but  is  no  more  than  a  theory.  It  seems  unques- 
tionable that  the  process  of  magnetization  is  attended  by 


56  NOTES   ON 


molecular  movements,  but  it  is  not  proved  that  magnetism 
is  due  to  molecular  currents.  Prof.  Hughes  has  recently 
experimented  on  tempered  steel,  using  an  ingenious  mod- 
ification of  the  induction  balance,  and  asserts  his  belief 
that  each  molecule  possesses  magnetic  polarity.  In  tem- 
pered steel  the  molecules  are  comparatively  fixed,  where- 
as in  soft  iron  they  possess  considerable  freedom  of  move- 
ment. He  thinks  the  process  of  magnetization  is  merely 
one  of  molecular  movement,  by  which  the  similar  poles  of 
the  molecules  are  brought  facing  the  same  way.  This 
final  position  is  retained  by  the  steel  but  lost  by  the  iron. 
Ampere  investigated  the  mutual  action  of  magnets  and 
currents  by  the  theory  of  action  at  a  distance  between 
currents,  but  if  in  Fig.  13  the  current  in  each  molecule  is 
supposed  to  have  one  line  of  force,  the  aggregation  of 
molecules  would  produce  the  large  number  of  lines  all 
passing  in  the  same  direction  that  the  magnet  is  found  to 
possess,  and  the  theory  of  lines  of  force  due  to  Faraday 
and  Maxwell  explains  all  magnetic  phenomena  as  well  if 
not  better  than  the  method  of  Ampere. 

43.  Best  Arrangement  of  Cells  (§  351). 

From  general  formula,  letting  w  be  the  total  number  of 
cells,  w  —  m  n 

mE  R  E      *  .     .     (i) 


mr 

,      r        R 

mr       R 

°4~  ./ 

t      —  -i  

4  

n 

n        m 

w        m 

*  Proof  without  calculus. 


In  equation  (i)  —  +    -  =  2    ,  / '  Rr    +    (    /  r.:r  _       /   R 
™          ™  V      «          \V    ^~        V     ™ 

This  is  a  minimum  when  the  square  it  contains  is  zero,  or  when 


in-r       mr 

In  that  case  R  —  — —  =  —    as  before. 
•w  n 


ELECTRICITY   AND    MAGNETISM.  57 

This  is  a  maximum  when  the  denominator  is  a  mini- 
mum. Differentiating  with  regard  to  m  and  making  first 
derivative  equal  zero. 

du         r         R  r         R  m^r        mr 

dm  ~~  iv     ~   m*  ~  w  ~  m'1  O1  w    ' '    n 

But  —  is  the  internal  resistance  of  the  battery.     Hence 
n 

the  rule,  that  the  best  arrangement  is  secured  when  the 
internal  resistance  of  the  battery  equals  the  external 
resistance  in  circuit. 

44.  Long  and  Short  Coil  Galvanometers  (§  352). 

In  the  use  of  a  galvanometer  it  is  desirable  that  it 
should  produce  a  readable  deflection  without  greatly  re- 
ducing the  current.  If  the  current  is  large  a  single  turn 
of  wire  will  cause  a  sufficiently  strong  field,  but  if  the  cur- 
rent is  small,  it  is  necessary  to  multiply  its  effect  by  pass- 
ing it  through  many  turns  in  order  to  obtain  a  good 
deflection. 

By  Note  37  the  field  at  the  centre  of  the  coil  is  — 

in  a  galvanometer  having  n>  and  -    -  in  another  having 

only  one  turn  around  its  needle.  The  resistance  of  the 
first  will  be  nearly  n  times  that  of  the  second.  Let  all 
the  resistance  in  the  circuit  external  to  the  galvanometer 
be  r,  and  g  be  the  resistance  of  the  galvanometer,  and  E 
the  E.  M.  F.,  supposed  constant.  Then 

C  =  — ^—     and  C  =  — 

r  +  g  r  +   ng 

\{r  is  small  C=  nC'  nearly,  or  the  current  is  reduced 
by  the  high  resistance  galvanometer  in  almost  the  same 


58  NOTES  ON 


ratio  that  the  field  is  increased.  There  is,  therefore,  no 
gain,  and  the  great  reduction  of  the  current  renders  the 
use  of  such  an  instrument  inadvisable.  If  r  is  large,  the 
resistance  is  not  increased  n  times  by  the  introduction  of 
ng  instead  of  g,  and  C  is  <nC',  or  the  field  is  strength- 
ened in  a  greater  ratio  than  the  current  is  decreased. 
There  is,  therefore,  a  gain  in  using  a  high  resistance  gal- 
vanometer. If  r.  is  very  large,  a  galvanometer  of  many 
turns  must  be  used  to  obtain  any  deflection  at  all. 

45.  Divided   Circuits  (§  353). 

It  is  at  first  difficult  to  understand  how  introducing 
another  resistance  in  parallel  circuit  can  reduce  the  total 
resistance,  but  it  is  to  be  recollected  that  the  current  pos- 
sesses a  greater  number  of  paths  to  flow  through.  By 
the  law  that  the  current  divides  proportionally  to  the  con- 
ductivities of  the  branches,  —  goes  through  one  branch 

and  —  through  the  other.     Let  R  be  a  resistance   such 

that  the  same  current  would  pass  through  it  as  through 
both  r  and  r1 '.  Then 


ill          _ 

-?;  =  —  +  -7    or  7?  = 


R       r       r'  r  +  r' 


r'  r 

But   -  is  less  than  r>  and  -  -    is  less  than  r.     The 


resistance  of  a  divided  circuit  is,  therefore,  always  less 
than  that  of  any  of  the  resistances  entering  into  it.  If 
there  are  three  conductors,  r,  r',  r",  we  have 


ELECTRICITY  AND   MAGNETISM.  59 


—  =  —  +  —,+  —  =  conductivity, 
R        r       r        .r" 

and  R  =  -    ,      "T       ,  „. 

rr   +  rr'    +  r  r1 

To  find  the  current  in  each  branch  :  Let  C  be  the  total 
current,  B  the  battery  resistance,  O  be  the  current  in  r', 
and  C"  that  in  r".  Then 


The  currents  are  inversely  as  the  resistances  through 
which  they  flow.  Taking,  therefore,  the  resistances  be- 
tween A  and  Bt  Figure  129, 

r'  r"  r" 


also      C:  C"  :  :  r"  :    r'  r"  „    or  C"  -C  -7-^-7,  .     .    (2) 

r'  +  r"  r'  +  r" 


46.  Shunts. 

These  formulas  are  of  great  importance  in  the  use  ot 
shunts  for  galvanometers.  If  a  current  is  so  powerful 
that  there  is  danger  of  its  injuring  the  galvanometer  coils, 
or  if  it  produces  a  deflection  too  near  90°,  the  galvanom- 
eter may  be  shunted  by  introducing  a  resistance  in 
parallel  circuit,  so  that  less  current  will  pass  through 
the  galvanometer. 
Letting  C  be  the 
current  when  the 
galvanometer  is  B 
unshunted,  C'  the 
total  current  when  Fiff-  *4- 

the  galvanometer  is  shunted,    Cg  and   Cs  the  currents 


60  NOTES  ON 


through  the  galvanometer  and  shunt  respectively,  G  the 
resistance  of  the  galvanometer,  S  that  of  the  shunt,  and  /? 
all  other  resistance, 

*  E 


But  as  — ~  is  less  than   G,  C'  is  greater  than   C,  or 

Lr     +     O 

the  introduction  of  the  shunt  has  increased  the  total  cur- 
rent in  circuit.  The  current  through  the  galvanometer  is 
from  (i),  Note  45, 

c  , 5^_    _          E  S 

> 

•     (3) 


j.\.     -r       — , 

G  + 
ES 


'  R  (G  +  S)  +  GS 

If  the  deflections  of  the  galvanometer  are  proportional 
to  the  currents, 


.-.  d  :  d'  :  :  R  (G  + 


These  formulas  may  be  simplified  if  -  ~—  -    be  called 

u.     This  proportion  is  sometimes  called  the  multiplying 
power  of  the  shunt.     By  its  use  (3),  the  current  through 

C' 

the  galvanometer  becomes  —  .      The    current   through 

the  shunt  is  C'  —•     The  resistance  of  the  shunted  gal- 

f~  *\  (^ 

variometer   -^  -  -    becomes     —  ,     and    the    resistance 
G  +  5  u 


ELECTRICITY   AND    MAGNETISM.  6 1 


necessary   to   add    to    the   circuit  to    retain    the    same 

total  current  is  G =  G  (U  ~      }.     This  ratio  u  is 

u  \     u     J 

that  between  the  sensibility  of  the  shunted  and  the  un- 
shunted  galvanometer.  Thus,  if  the  resistance  of  a  shunt 
which  will  reduce  the  sensibility  of  a  galvanometer  of 
1,000  ohms  one  hundred  times  is  required, 

looo  +  S        _        1000  i    ~ 

U  =    IOO  = ~ .'.  O   —     •=    I.OI    =  (jr. 

S  99  99 

If  the  current  is  kept  the  same  by  adding  a  resistance  of 

G   (  —       -  ),  C—  C,  and  —  will  pass  through  the  galva- 

\     u     J  u 

nometer. 

47.  Kirchhoff's  Laws  (§353). 

The  application  of  theae  useful  laws  may  be  illustrated 

by  the  figure.     The  fol- B 

lowing    equations    are 

derivable.      From    the 

first  law,  that   in   any   b- 

network  of  wires   the 

algebraic   sum   of  the 

currents  meeting  at  a        v 

point  is  zero.  Fig.  15. 


I 


A 


At  A        c  =  Ci  +  c^ 
MB        c^  +  c^  —  c. 

From  the  second  law,  that  in  any  closed  circuit  the 
electromotive  force  is  equal  to  the  sum  of  the  separate 
resistances,  each  multiplied  by  the  strength  of  current 
flowing  through  it. 

In  the  left  hand  circuit  c  (r  +  b]  +  cl  rl  =  E. 

In  the  right  hand  circuit  r,  rl  —  c-ir*  •=  zero. 


62 


NOTES   ON 


48.   Fall  of  Potential  (§  357). 

Let  A  and  B  be  the  poles  of  a  battery.  They  will  have 
different  potentials,  numerically  equal  but  of  opposite 
signs,  and  the  battery  may  be  considered  to  preserve  a 
constant  difference  of  potential  between  them.  Connect 
the  points  A  and  B  through  an  external  resistance.  The 

p  potential  will  fall 
along  this  resistance, 
and  it  is  required  to 
find  the  potential  at 
any  point. 

Draw  A  B  to  rep- 
resent in  length   the 
B  value  of  the  external 
resistance,  and  at  A 
and  B  erect  perpen- 
diculars, one  positive 
and  the  other  nega- 
tive, to  represent  pro- 
r- l6-  portionally  the  poten- 

tials at  these  points. 

From  Kirchhoff 's  second  law,  in  any  part  of  a  circuit 
of  resistance  r,  E  —  C  r.  In  another  part  of  resistance 
r',  E  =  C'  r1 ,  E  and  E'  representing  the  difference  of 
potentials  between  the  ends  of  the  portions  of  the  circuit 
considered.  If  the  resistances  are  both  in  the  same  cir- 
cuit C  =  C',  hence  E  \  E'  :  :  r  :  r',  or  the  differences  of 
potential  in  the  same  circuit  are  proportional  to  the  resist- 
ances through  which  they  act.  Assuming  one  potential 
to  be  zero,  the  potentials  at  the  other  points  vary  directly 
as  the  resistances  separating  them  from  the  point  of  zero 
potential,  and  as  in  the  figure  the  horizontal  line  repre- 
sents resistance  and  the  ordinate  BH  potential,  it  is  evi- 


ELECTRICITY  AND   MAGNETISM.  63 

dent  that  ordinates  at  other  points  cut  off  by  the  line  CH 
will  correctly  represent  the  potential  at  these  points,  as 
they  and  they  only  will  satisfy  the  above  proportion. 
The  line  CH  represents,  therefore,  the  fall  of  potential  in 
the  resistance  BC.  If  CH\s  prolonged  to  K  the  triangles 
I~4C  and  CHB  are  similar,  and  since  by  hypothesis 
A  K  is  equal  to  BH,  AC  =  BC,  or  C,  the  point  of  zero 
potential,  is  midway  between  the  poles,  and  the  line  HK 
represents  the  fall  of  potential  along  the  resistance  AB. 
It  is  to  be  noticed  that  in  the  figure  the  difference  or 
potential  between  A  and  B  is  BH  +  AK  \  or  if  —  Fis  the 
potential  at  A  and  +  V  at  B,  the  difference  of  potential  is 
V  -  (-  F)  =2  V.  The  difference  between  A  and  D  is 
v  —  (—  V]  =  V  +  v.  If  the  point  C  is  connected  with 
the  earth,  as  it  is  already  at  zero  potential,  the  potential 
is  not  changed  anywhere  in  the  circuit.  If,  however, 
another  point  Dt  whose  potential  is  +  vt  is  connected  to 
earth  its  potential  is  lowered  by  the  amount  v,  and  as  the 
battery  preserves  a  constant  difference  of  potential  be- 
tween A  and  B,  the  absolute  potential  of  all  points  in 
the  circuit  is  also  lowered  by  the  amount  v.  The  fall 
of  potential,  is,  therefore,  the  same,  and  is  represented  by 
drawing  a  line  M  N  through  D  parallel  to  HK,  and  the 
potential  at  any  point  is  the  length  of  the  ordinate  at  that 
point  intercepted  byMN. 

The  difference  of  potential  between  A  and  B  is  now 
BN+ AM= 

(  F  -  -z/ )  +  (  F  +  ?/ )  =  2  Fas  before. 

If  the  negative  pole  A  of  the  battery  is  connected  to 
earth,  the  potential  of  all  points  of  the  circuit  is  raised 
by  the  amount  V,  and  the  line  of  potential  assumes  the 
position  A  P.  The  potential  of  every  point  of  the  circuit 


NOTES   ON 


is  now  positive,  but  the  differences  of  potential  are  the 
same  as  at  first. 

49.  Wheatstone's  Bridge  (§  358). 

Connect  the  poles  of  a  battery  by  two  resistances,  P  Q 
2 


Fig.  17. 

and  PQ',  and  at  P  erect  a  perpendicular  to  represent  the 
difference  of  potential  due  to  the  battery.  Then  the  lines 
ZQ  and  ZQ  will  represent  the  fall  of  potential  in  the 
resistances  PQ  and  PQ'.  The  potential  at  any  point  on 
QQ'  being  the  ordinate  cut  off  by  the  lines  ZQ  or  ZQ,  if 
a  galvanometer  be  joined  to  two  points  TV7  and  M  at  which 
the  ordinates  A^Tand  MY  are  equal  there  will  be  no  de- 
flection of  the  needle.  But  from  the  figure,  since  XY'is 
parallel  to  QQ,  the  triangles  ZRX and  XNQ  are  similar 
and  NQ  :  RX  :  :  A7X  :  RZ  ; 

RY::MY:RZ\    but  NX  =  MY 

RX:  :  MQ  :  RY, 

B     -.:€:£>, 


also 


or 


MQ 
.NQ 

A 


as  in  Fig.  130  and  %  358.     When,  therefore,  a  galvanom- 
eter joined  to  the  junctions  of  two   pair   of  resistances 


ELECTRICITY  AND   MAGNETISM. 


through  which  a  current  is  flowing  shows  no  deflection, 
the  resistances  are  proportional  to  each  other. 

An  important  fact  somewhat  difficult  to  understand  at 
first  is  apparent  from  Fig.  17.  The  resistances  PNQ  and 
PMQ'  (see  also  Fig.  130)  are  unequal,  while  having  the 
same  electromotive  force  acting  in  each.  The  currents 
in  the  two  branches  are  therefore  unequal.  Beginners 
are  liable  to  regard  the  balance  in  Wheatstone's  Bridge  as 
due  to  an  equality  of  currents  ;  but  this  is  wrong,  the 
equality  of  potentials  at  the  galvanometer  terminals  being 
the  condition  of  balance.  This  is  equivalent  to  saying  that 
there  is  no  current  through  the  galvanometer,  offering 
another  method  of  proof  as  follows  : 

50.  Proof  of  Theory  of  Wheatstone's  Bridge  by  Kirch- 
hoff's  Laws. 

Let  the  currents  in  the  different  branches  be  represented  by 
A 


Fig.  18. 

cz,  etc.,  and  the   corresponding  resistances   by  r,  rlf 
5 


66  NOTES    ON 


etc.  Let  j£ be  the  E.  M.  F.  and  b  the  resistance  of  the  battery. 
From  the  first  law 

At    A  f  i  /    \ 

•**•*•    •*•*  ^2    • —    ^3      i      C '5       .         .        .          ..         .        .         .  (I) 

B           c     =  fa   +  d (2) 

C*           ^4  =  c\   +  c<> (3) 

"  D          c    =  c*  +  c, (4) 

By  the  tin  ABC,  ca  r3  -•  ct  rt  —  c*,  G  =  0       .     .     .     .  (5) 

second  j  "  ADC,   c*  r-2  +  c<,  G  —  d  ri  =  0       .     .     .     .  (6) 

law,     (  "  BbDC,  c  (b  -f  r\  +  Cl  ^  +  ct  rt  -  E  -      O.  (7) 

Adjust  ri,r<i,r3  and  r^  until  the  galvanometer  shows  no 
deflection,  then  c$  is  zero.  The  above  equations  being  general, 
substitute  zero  for  d,  and  they  become 

(i)  ci  =  ca  (5)     ca  ra  =  d  r4      or  c2  r3  —  c 4  rt     .       (8) 

(3)  d  =  fi  (6)     c*  r*  =  c,  r,      or  c,  r,  =  c,  r,     .       (9) 

Dividing  (8)  by  (9)  -  =  -  . 

When,  therefore,  the  galvanometer  shows  no  deflection,  the 
arms  of  the  bridge  are  proportional.  The  accuracy  of  the  meas- 
urement depends,  of  course,  on  the  sensibility  of  the  galvanom- 
eter. 

51.  Measurement  of  Electromotive-Force  (§  360).* 

(a)  WHEATSTONE'S  METHOD. 

Let  G  be  the  resistance  of  the  galvanometer,  R  the  remaining 
resistance  in  circuit,  and  p  the  resistance  necessary  to  add  to 
the  second  batteiy  to  reduce  its  deflection  to  that  of  the  first. 
Then  for  a  deflection  d 

*  The  memoranda  of  Notes  51,  52  and  53  are  confined  strictly  to  an  ex- 
planation of  the  methods  of  measurements  referred  to  in  "  Thompson's 
Elementary  Lessons,"  §  360,  361  and  362  Other  methods  may  be  obtained 
from  any  work  on  electrical  measurements. 


ELECTRICITY  AND   MAGNETISM. 


E 

E' 

R  +  G 

R  - 

\-  p  H 

-  G 

whence 

R  4-   G 

R  - 

\-  P  J 

-  G 

(i) 

To  bring  the  deflection   to  d' ,  extra  resistances  have  to  be 
added.     Represent  these  by  r  and  r1 .     For  the  deflection  d1 


, 
whence  - 


R  +  G  +  r 
R  +  G 


R  +  p  +  G  +  r1 


r' 
- 


R  +  p   +  G 


.     .     (2) 


Substituting  (ij  in  (2)  it  reduces  to 


-  =  — ; ;   or,  r'  :  r  :  :  E'  :  E. 

(t>)  CLARK'S  METHOD. 

Clark's  method  requires  three  cells.     E  furnishes  a  current 
and  has  the  highest 
E.    M.    F.,  E'   is    a  \       C 

standard  cell,  gener- 
ally that  of  Clark 
(§  177),  and  £v/isthe 
cell  to  be  tested.  If 
E"  has  a  higher 
E.  M.  F.  than  1.457 
two  or  more  Clark's 
cells  must  be  used 
at  E' '.  The  similar 
poles  of  the  cells 
are  connected  to 
A.  In  measuring, 


Fig.  19. 


E"  is   first  disconnected  and  the  needle  of    G'  is  brought  to 
zero  by  adjusting  P.     E"  is  then  connected  at  A  and  the  slid- 


68  NOTES   ON 


ing  piece  C  is  moved  along  the  resistance  A  B,  shifting  contact 
until  G"  also  shows  no  deflection. 

From  Kirchhoff's  laws  : 

At  A  c  +  c'  +  c"  —  K  —  o     .     .     .     .  (i) 

"   C  K-c"  -  K'         =  o    .     .     .     .  (2) 

"  B  K'  -  c'  -c            =o     ....  (3) 

In  circuit  G'AB  c'  (r>  +   G'}  +  Ka  +  A"b  -  E'  -  o     .  (4) 

A  G"C  c"  (r"  +  G")  +  Ka  -  E"  =  o     .  (5) 

But  when  adjustment  is  secured  c'  and  c"  are  each  zero. 
Substituting  these  values 

from  (\)c  —  K    from  (2)  K  -  K'     from  (3)  c  =  K' 

from  (4)  Ka  +  K'b  =  £'  ;    or,  K  (a  +  b)  =  E'      .     .     .     (7) 

from  (5)  Ka  -  E"     . (8) 

Combining  (7)  and  (8) 

E'  :  E"  :  :  a  +  b  :  a. 

(C)  QUADRANT    ELECTROMETER. 

The  two  poles  of  a  standard  cell  are  connected  to  the  quad- 
rants, the  same  pole  being  in  connection  with  opposite  seg- 
ments, as  I  and  3,  Fig.  101.  The  deflection  of  the  needle  is 
then  noted.  The  standard  cell  is  then  disconnected  and  the 
one  to  be  tested  substituted  in  the  same  way.  From  the  ratio 
of  the  deflections,  the  ratio  of  the  electro-motive  forces  may  be 
obtained.  Care  must  be  taken  that  the  needle  is  electrified  to 
the  same  potential  in  the  two  measurements. 

52.  Measurements  of  Internal  Resistance  (§  361). 

(a)  Connect  the  battery  in  circuit  with  a  galvanometer  and  a 
box  of  resistance  coils,  the  resistances  being  B,  G  and  R  re- 
spectively. Note  the  deflection  d  in  the  galvanometer.  In- 
crease R  to  R'  and  note  the  deflection  d' .  Then  if  a  tangent 
galvanometer  is  used 


ELECTRICITY   AND    MAGNETISM. 


69 


tan  d:ia.nd' 


Reducing, 


Jj    = 


R'  tan  d'  —  R  tan  d 

-  :  -  r— 

tan  d  —  tan  ^ 


_ 

—    Cr. 


If  G  is  of  no  resistance,  and  the  first  deflection  was  taken 
with  no  other  appreciable  resistance  than  that  of  the  battery  in 
circuit 

_        R'  *an</' 

tan  d  —  tan  d '  ' 

(c)  MANGE'S  METHOD. 

The  cell  whose  resistance  is  to  be  measured  is  placed  in  the 
bridge  as  an  unknown  resistance,  and  a  galvanometer  and  key 
(not  in  the  same  branch)  are  connected  as  in  the  figure.  The 
arrows  denoting  the  direction 
of  the  current,  it  is  at  once 
evident  that  a  current  passes 
through  the  galvanometer  all 
the  time,  and  continues  to  do 
so  unless  the  resistance  in  c 
and  d  becomes  zero.  Every 
change  of  resistance  in  a,  c 
or  d  will  affect  the  current 
flowing  through  G,  as  will  also 
the  opening  or  closing  of  the 
key  k  when  any  difference  of 
potential  exists  between  A  and  B.  If  on  pressing  k  there  is  no 
change  in  the  galvanometer  deflection,  it  follows  that  A  and  B 
are  at  the  same  potential  and  that  consequently 

a  :  b  ::  c  :  d  or  b  —  —  , 

c 

the  common  relation  of  the  Wheatstone  bridge.  If  the  galva- 
nometer is  a  sensitive  one,  the  deflection  will  be  nearly  90°  and  as 
in  that  position  a  change  of  current  produces  but  little  effect  on 


Fig.    20. 


70 


NOTES  ON 


the  needle  it  is  necessary  to  reduce  the  deflection  either  by  the 
use  of  a  magnet  or  by  shunting  the  galvanometer. 

53.  Measurement  of  the  Capacity  of  a  Condenser  (§  362). 

(a)  Let  V  =  the  known  potential, 
x  —  the  unknown  capacity, 
C  —  the  capacity  of  the  standard  condenser. 

The  original  charge  in  the  condenser  of  unknown  capacity 
is  Vx.  On  connecting  the  standard  condenser  the  capacity  is 
increased,  and  the  quantity  being  the  same  the  potential  falls 
to  V" 

.'.  Vx  =  V  (x  +  C\ 
and  x  :  x  +  C  :  :  V  :  V. 

(8)  The  impulse  acting  to  deflect  the  needle  varies  as  the 
quantity  of  electricity  passing.  If  the  two  condensers  are 
charged  from  the  same  cell  the  potential  is  equal.  Conse- 
quently (Note  5), 

sini</:  sin**/'  :  :  VC  :  W 
::  C:  C  . 


(c)  There  is  apparently  some  error  in  stating  this  method  in 
"Thompson."  It  probably  refers  to  a  common  and  quite  ac- 
curate method  applied  by  Sir  Wm.  Thomson  to  the  measure- 
ment of  the  capacity  of  cables. 

If  the  poles  of  a  battery  are 
connected  through  a  high  resist- 
ance I?i  +  J?  and  a  point  F  is 
connected  with  earth,  its  poten- 
tial becomes  zero  and  points  on 
either  side  separated  from  it  by 
equal  resistances  are  at  equal  but 
opposite  potentials.  If  the  re- 
sistances are  unequal, 


ELECTRICITY  AND   MAGNETISM.  7 1 


Potential  at  A  :  Potential  at  C  :  :  RI  :  R, 
or         V,  :  F2  : :  R,  :  R, 

•••/•=¥• 

Charge  the  t\vo  condensers  by  making  contact  on  opposite 
sides  of  F  at  such  points  that  their  charges  just  neutralize.  If 
A  and  C  are  the  points, 


(</)  Charge  the  condenser  and  discharge  it  through  a  galva- 
nometer, noting  the  deflection  ;  charge  it  again  to  the  original 
potential  and  allow  it  to  discharge  slowly  through  a  very  high 
resistance.  After  discharging  a  definite  time  T,  note  the  deflec- 
tion it  will  give  when  connected  directly  to  the  galvanometer. 
Then 

T 
Capacity  =  -  —  ; 

2.303^  log  — 

V  and  v  are  potentials,  but  as  the  ratio  only  is  needed,  this  can 
be  obtained  from  the  ratio  of  the  two  deflections. 

54.  Determination  of  the  Ohm  (§  364). 

If  a  coil  is  rotated  in  a  magnetic  field  a  current  is  in- 
duced, which  deflects  a  needle  at  the  centre  of  the  coil. 
The  force  exerted  by  this  current  may  be  shown  to  be 

,  and  the  moment  acting  on   a  magnet  deflected 
%K?JK. 

L?  VH 

through  an  angle  0  is  ^j-^ —  ml  cos  0,  where  L  is  the 
^•K  1\ 

length  of  the  coil,  Fthe  velocity  of  rotation,  H the  inten- 
sity of  the  field,  K  the  radius  of  the  coil  and  R  its  resist- 
ance. The  moment  of  the  force  of  the  earth's  magnetism 


72  NOTES   ON 


tending  to  bring  the  deflected  magnet  into  the  meridian 
is  Hml  sin  6.  Equating  these  moments  when  the  needle 
maintains  a  steady  deflection, 


ml  cos  0  =  Hml  sin  6  ; 


The  value  of  R  is  given  in  absolute  units  of  resistance. 

55.  Practical    Electromagnetic    Uuits   of    Heat  (§  367) 
and  of  Work  (§  378). 

As  shown  in  Note  7,  the  measure  of  the  work  done  in 
moving  a  quantity  of  electricity  is  the  product  of  the 
quantity  of  electricity  by  the  difference  of  potential 
through  which  it  is  moved,  or 

Work  =  Quantity  ( V,  -  F2)  =  Ct  x  E. 

If  the  current,  time,  and  electromotive  force  are  all 
expressed  in  absolute  units,  the  work  is  given  in  ergs. 
Substituting  for  the  practical  units  of  current  and  elec- 
tromotive force,  the  ampere  and  the  volt,  their  values  in 
absolute  units, 

Work  =  io  —  *  x   io8  =  io7  ergs  per  second. 

This  equation  gives  a  practical  unit  of  work,  correspond- 
ing to  the  practical  units  of  current  and  electromotive 
force  referred  to,  and  Dr.  Siemens  has  proposed  that  it 
be  called  the  "watt"  This  suggestion  has  been  well 
received,  and  the  unit  is  coming  into  use  in  electrical 
calculations.  The  value  of  the  watt  is  io7  ergs,  and  it 
may  be  defined  as  the  work  done  by  a  current  of  one  am- 


ELECTRICITY   AND   MAGNETISM.  73 

pere  in  a  portion  of  the  circuit  in  which  the  potential  falls 
one  volt. 

I  H.  P.  English  =  33000  foot-pounds  per  minute. 
=      550     "         "         "     second. 
=  76.04  kilogrammetres  per  second. 
==  76.04  x   io5    gramme-centimetres  per 

second. 

z=  76.04  x  981   x  10*  ergs  per  second. 
=      746  x   io7  ergs  =•  746  watts. 

To  find,  therefore,  the  work  done  by  an  electric  cur- 
rent in  any  portion  of  a  circuit,  measure  the  differ- 
ence of  potential  in  volts  between  the  ends  of  the  por- 
tion considered,  multiply  it  by  the  current  in  amperes, 
and  divide  by  746.  The  quotient  is  the  work  in  horse- 
power. In  the  above  calculation  the  dimensional  equation 
of  work  cannot  be  used  for  the  change  of  foot-pounds  to 
kilogrammetres,  as  these  are  statical  units,  and  the  dimen- 
sions are  for  dynamical  units  only. 

As  heat  and  work  are  the  same,  the  heat  measured 
in  ergs  given  off  in  any  part  of  a  circuit  is  also  CEt,  or 
substituting  the  value  of  E  from  Ohm's  Law, 

Heat  in  ergs  —  CzRt. 

This  formula  is  more  generally  used,  as  it  gives  the 
amount  of  heat  developed  in  a  resistance.  Substituting 
as  before  the  values  of  the  ampere  and  ohm  in  absolute 
units,  the  heat  is  measured  in  practical  units,  each  of 
which  is  of  the  value  of  io7  ergs.  This  unit  is  some- 
times called  the  "joule,"  and  may  be  defined  as  the  heat 
evolved  in  one  second  by  a  current  of  one  ampere  in  a 
resistance  of  one  ohm. 

To  obtain  the  value  of  the  joule  in  terms  of  the  more 


74  NOTES   ON 


common  thermal  unit,  which  is  the  amount  of  heat  nec- 
essary to  raise  one  gramme  of  water  one  degree  centi- 
grade, it  is  necessary  to  use  the  mechanical  equivalent  of 
heat,  which  is  42,400  gramme-centimetres.  A  gramme  of 
water  in  falling  42,400  centimetres  acquires  sufficient 
energy  to  raise  its  temperature  one  degree  centigrade  if 
suddenly  stopped.  The  thermal  unit,  therefore,  equals 
42,400  gramme-centimetres  or  42,400  x  981  ergs  =  4.16 
x  io7  ergs,  or  approximately  4.2  x  io7  erg's.  But  as 
the  joule  is  io7  ergs,  the  water-gramme-degree  heat  unit  is 
equal  to  4.2  joules,  or  the  joule  is  .238  of  the  thermal 
unit  referred  to. 

The  watt  and  joule  are  of  the  same  value,  but  one  ex- 
presses the  energy  given  off  in  a  circuit  in  terms  of  power, 
and  the  other  in  heat. 


ELECTRICITY   AND    MAGNETISM.  75 


V.  ELECTRIC   LIGHTS. 
56.    The    Voltaic   Arc    (§  371). 

If  a  powerful  electric  current  is  broken  at  any  point, 
there  is  a  bright  spark  at  the  break,  and  if  the  two  termi- 
nals of  the  circuit  on  each  side  of  the  break  are  kept  at  a  con- 
stant short  distance,  a  steady  light  or  arc  will  be  produced. 
The  color  of  the  light  varies  with  the  materials  between 
which  the  arc  is  formed.  The  heat  produced  is  the  highest 
known,  and  most  substances  fuse  so  readily  in  the  arc 
that  they  cannot  be  used  for  electrodes.  For  this  reason 
gas  carbon,  which  is  practically  infusible  and  of  compar- 
atively low  electrical  resistance,  is  universally  used  for  the 
pencils  of  arc  lights.  If  the  image  of  the  arc  is  thrown 
on  a  screen  the  greater  part  of  the  light  is  seen  to  be  due 
to  the  carbon  points  being  heated  white  hot,  the  arc  itself 
being  generally  bluish  and  less  brilliant.  If  a  magnet  is 
brought  near  the  arc,  the  interaction  of  magnets  and 
currents  is  well  illustrated  by  the  movements  of  the  arc 
to  one  side,  and  it  is  even  possible  to  deflect  it  until  it 
assumes  the  position  of  a  blowpipe  flame. 

The  electric  light  was  prevented  from  coming  into  every- 
day use,  so  long  as  the  current  it  required  could  not  be 
obtained  from  any  cheaper  source  than  the  voltaic  battery. 
The  invention  of  dynamo  machines,  by  greatly  decreasing 
the  cost  of  electric  energy,  rendered  the  extensive  use  of 
the  arc  light  a  possibility.  This  may  be  best  illustrated 
by  an  example. 

A  Number  7  Brush  machine  works  sixteen  arc  lights  in 


NOTES   ON 


series.  The  E.  M,  F.  of  the  circuit  is  839  volts,  the  inter- 
nal resistance  of  the  machine  10.5  ohms,  the  resistance  of 
the  lights  and  leading  wires  73  ohms  and  the  current  10 
amperes.  Assuming  the  electromotive  force  of  a  Grove 

2 

cell  to  be  1.8  volts  and  its  internal  resistance  — ,  thenum- 

10 

her  of  cells  necessary  to  do  the  work  of  the  machine  may 
be  calculated. 

To  obtain  the  E.  M.  F.  the  number  of  cells  necessary  is 

— |  =  466  in  series.  But  466  cells  in  series  have  an  inter- 
I.o 

nal  resistance  of  93.2  ohms,  and  that  the  current  may  be 
the  same,  enough  cells  must  be  introduced  in  arc  to  re- 
duce the  battery  resistance  to  that  of  the  machine. 

10.5 

is  nearly  nine,  and  it  is  therefore  necessary  to  use  466 
groups  in  series,  each  group  containing  9  cells  in  arc  or 
4,194  cells  in  all.  This  number  of  cells  would  cost  ten 
times  as  much  as  the  machine,  and  could  not  keep  up  the 
current  for  more  than  two  or  three  hours. 

Calculating  in  the  same  way  with  a  constant  battery, 
assuming  the  E.  M.  F.  of  a  gravity  cell  to  be  1.08  volts 
and  its  internal  resistance  5  ohms,  it  will  be  found  neces- 
sary to  arrange  777  groups  in  series,  each  group  contain- 
ing 369  cells  in  parallel  arc,  a  total  of  286,713  cells,  a 
greater  number  in  all  probability  than  are  in  use  in  the 
United  States.  The  invention  of  the  dynamo  machine 
has  therefore  not  only  operated  to  diminish  the  cost  of  the 
electric  light,  but  to  bring  it  within  the  bounds  of  com- 
mercial practicability. 

As  a  general  rule,  the  light  given  out  by  an  arc  varies 
as  the  current.  A  small  current  will  give  no  light  at  all, 
and,  as  stated  in  §  371,  a  certain  electromotive  force  and 
current  are  necessary  for  the  production  of  a  satisfactory 


ELECTRICITY   AND    MAGNETISM. 


77 


light.     After  that  point  a  stronger  current  causes  more 
light. 

57.  Arc  Lamps  (§  372). 

Gas  carbon  pencils  are  now  used  almost  exclusively,  and  are 
frequently  coated  with  a  thin  film  of  copper  to  prevent  the 
oxidation  and  waste  of  the  carbon  before  it  becomes  incandes- 
cent. Two  general  methods  are  in  use  for  regulating  the  dis- 
tance of  the  carbons  from  each  other,  one  using  clock-work  and 
the  other  regulating  directly  by  the  current.  The  most  widely 
used  of  the  latter  class  is  the  Brush  lamp,  a  plan  of  which  is 
given  in  the  figure.  The  current  entering  at  A,  divides  at  B 
into  two  branches  which  pass 
around  the  bobbin  C  in  oppo- 
site directions,  one  branch  be- 
ing a  coarse  wire  of  low  resist- 
ance and  in  the  same  circuit 
as  the  carbons,  and  the  other 
branch  S  S  being  a  shunt  of 
high  resistance  to  the  lamp, 
connecting  the  terminals  D  and 
G.  Inside  the  bobbin  is  a  soft 
iron  armature  F,  which  is  at- 
tached to  the  upper  carbon. 
When  a  current  passes  the  two 
branch  circuits  on  the  bobbin 
C  tend  to  magnetize  it  in  oppo- 
site directions,  but  the  resist- 
ances and  number  of  turns  in 
the  two  circuits  are  so  propor- 
tioned that  the  magnetic  field  due  to  the  low  resistance  branch 
is  the  stronger,  and  the  armature  F  is  therefore  attracted,  lift- 
ing the  upper  carbon  and  establishing  the  arc.  Should  the 
carbons  become  too  widely  separated  the  resistance  of  the  arc 
and  consequently  of  the  coarse  wire  circuit  on  C  increases, 
diminishing  the  current  in  C  and  increasing  that  in  the  shunt  S. 


NOTES   ON 


The  field  due  to  the  shunt  is  therefore  strengthened  and  that 
due  to  the  coarse  wire  diminished,  allowing  the  armature  F  to 
fall  slightly,  bringing  the  carbons  nearer  together.  By  the  de- 
vice of  the  two  opposing  fields,  due  to  the  coils  on  C  being 
wound  in  opposite  directions,  the  feeding  of  the  lamp  is  there- 
fore done  automatically,  and  the  actual  distance  of  the  two  car- 
bons varies  but  little.  In  the  lamp  as  constructed,  two  bobbins 
are  used  in  parallel  arc,  and  the  armature  F  clutches  the  upper 
carbon.  A  plunger  moving  in  glycerine  is  also  attached  to  the 
upper  carbon  to  render  the  movements  less  sudden,  and  the 
shunt  circuit  S  S  passes  around  another  bobbin,  which,  by  at- 
tracting an  armature,  closes  the  main  circuit,  and  short  circuits 
the  lamp  in  case  the  carbons  are  broken  or  the  adjustment 
does  not  work.  The  figure  is  designed  merely  to  illustrate  the 
general  method. 

The  Foucault  regulator  shown  in  Fig.  138  is  more  com- 
plicated than  the  Brush.  The  two  carbons  are  clamped  to 
rods  which  are  moved  by  clock-work,  the  gearing  being  such 
that  the  positive  carbon  moves  twice  as  rapidly  as  the  negative. 
In  this  way  the  arc  is  kept  approximately  in  the  same  position, 
admitting  of  focussing  in  a  projecting  apparatus.  The  clock- 
work consists  mainly  of  two  barrels  driven  by  springs  and  act- 
ing through  gearing  on  the  rods  carrying  the  carbons,  one  bar- 
rel separating  the  carbons  and  the  other  bringing  them  nearer 
together.  The  electromagnet  seen  in  the  base  of  Fig.  138  is 
in  the  main  circuit,  and  its  armature  works  a  system  of  levers, 
the  last  one  of  which  acts  by  a  pawl  on  two  small  flies,  each 
connected  through  the  gearing  with  one  of  the  barrels.  If,  then, 
the  current  becomes  too  strong,  the  armaiure  is  attracted,  one  of 
the  fiies  is  released  and  the  corresponding  barrel  sets  the  clock- 
work in  operation,  separating  the  carbons  slightly.  As  they 
approach  their  normal  distance  apart  the  current  diminishes,  and 
the  armature  moves  away  slightly,  pawling  the  fly.  As  the  car- 
bons burn  away,  the  current  is  diminished  still  more,  the  arma- 
ture is  less  attracted,  the  other  fly  is  released  and  the  clock- 
work moves  to  bring  the  carbons  nearer  together.  The  arc 


ELECTRICITY   AND    MAGNETISM.  79 

may  be  formed  at  any  height  by  moving  the  upper  carbon  by 
means  of  the  rod  at  the  top  of  the  regulator. 

58.  Incandescent  Lamps  (§  374). 

That  electric  lighting  may  be  of  universal  utility,  lamps 
are  necessary  which  give  only  a  moderate  light.  Arc 
lights  cannot  be  made  to  work  with  certainty  and  econ- 
omy at  low  illuminating  power,  and  the  incandescent 
lamp  is  therefore  coming  into  use.  In  this  case  the  light 
is  given  off  from  a  portion  of  the  circuit  which  is  heated 
white  hot  by  the  passage  of  the  current,  and  this  requires 
that  that  portion  shall  be  of  high  resistance  (g  367)  and 
practically  infusible.  After  long  experimenting,  carbon 
has  been  fixed  upon  as  the  best  material,  and  it  is  now 
used  in  all  types  of  incandescent  lamps.  A  small  fibre  of. 
carbon  is  obtained  by  heating  some  vegetable  substance 
out  of  contact  with  the  air  and  driving  off  all  volatile 
matter.  Edison  uses  bamboo  fibre,  Swan  cotton  thread, 
Lane-Fox  a  grass  fibre,  and  Maxim  paper.  This  fibre  is 
mounted  in  a  vacuum,  on  the  perfection  of  which  depends 
greatly  the  time  the  lamp  will  last,  the  presence  of  a  small 
amount  of  oxygen  insuring  the  destruction  of  the  fibre  by 
chemical  action.  Even  in  a  perfect  vacuum  the  fibre  will 
eventually  give  way  on  account  of  what  is  known  as  the 
"  Crookes's  effect,"  a  transfer  of  molecules  of  carbon  across 
from  one  heel  of  the  carbon  to  the  other.  Alternate  cur- 
rents by  wearing  each  heel  away  equally  tend  to  lengthen 
the  lifetime  of  the  lamp,  but  in  every  case  the  final  giving 
way  of  the  fibre  is  a  matter  of  time  only.  The  lifetime 
may  be  greatly  prolonged  by  working  the  lamp  below  its 
normal  power.  The  life  of  an  Edison  lamp,  working  with 
its  normal  current,  is  now  (1883)  probably  about  1,000 
hours. 


8o  NOTES   ON 


As  pointed  out,  the  condition  that  much  heat  should  be 
developed  in  any  part  of  a  circuit  is,  that  that  part  should 
be  of  high  resistance.  If  several  lamps  were  piaced  in 
series,  the  resistance  of  the  circuit  would  be  so  great  that 
the  current  would  be  insufficient,  and  they  are,  therefore, 
placed  generally  in  parallel  arc,  reducing  the  external  re- 
sistance to  such  an  extent  that  each  lamp  has  it  full  share 
of  current.  Assume  a  battery  or  machine  giving  an  elec- 
tromotive force  of  200  volts  at  its  terminals  and  having  an 
internal  resistance  of  five  ohms.  If  two  lamps  each  of  100 
ohms  resistance,  and  requiring  one  ampere  to  give  their 
normal  light,  were  placed  in  the  exterior  circuit  in  series, 

the  current  would  be =  .075  amperes — not  suffi- 

200+  5 

cient  to  work  the  lamps  at  their  normal  standard.  By 
placing  the  lamps,  however,  in  parallel  arc  the  same  ma- 
chine would,  under  the  same  conditions,  work  twenty 

lamps,  thus,  • —  =.-  20  amperes,  or  one  in  each  lamp. 

l-i+s 

That  all  the  lamps  may  give  the  same  light,  the  resistances 
must  be  equal,  otherwise  some  will  have  stronger  currents 
passing  through  them  than  others.  Placing  a  larger  num- 
ber in  parallel  arc  will  still  further  reduce  the  resistance, 
strengthening  the  current,  but  increasing  the  number  of 
parts  into  which  the  current  must  divide.  Thus  in  the 
case  above,  the  machine  cannot  work  21  lamps,  for 

=  20.47  amperes,  or  only  .975  to  each  lamp. 


A  most  important  feature  in  incandescent  lighting  is  the 
change  of  resistance  of  carbon  by  temperature.  The  re- 
sistance of  metals  increases  with  the  current  passed 
through  them,  while  that  of  carbon  decreases,  the  reduc- 


ELECTRICITY   AND   MAGNETISM. 


8l 


tion  in  an  incandescent  lamp  being  between  40  and  50 
per  cent.  As  the  good  working  of  any  system  of  lighting 
depends  on  a  correct  adjustment  of  resistances,  this  pecu- 
liarity of  carbon  must  be  allowed  for.  The  following 
measurements  made  on  an  Edison  lamp  at  the  U.  S.  Tor- 
pedo Station,  Newport,  illustrate  the  change  of  light  and 
of  resistance  accompanying  a  change  of  current. 


Current. 

Resistance. 

Candles. 

.000 

134- 

Lamp  cold. 

.114 

H0.5 

Cherry  red. 

.203 

94-7 

Bright  red. 

.309 

86.7 

Orange. 

.440 

8l.8 

1.8  Candles. 

.680 

735 

7.2        " 

.750 

72.8 

11.4 

.810 

70.6 

16.4 

.890 

69.9 

2I.O          " 

It  is  seen  that  a  current  of  to"  °f  an  ampere  produced  a 
light  just  perceptible,  but  that  every  increase  after  that 
produced  a  much  greater  proportionate  increase  of  light. 
The  lamps  are,  therefore,  more  economical  of  energy  the 
more  light  they  give,  but  working  with  high  currents  in- 
sures the  rapid  destruction  of  the  lamp.  At  present  the 
Edison  lamps  are  the  most  economical,  yielding  under 
good  working  conditions  about  twelve  sixteen-candle 
lamps  per  horse  power  of  current  (See  3  378). 
6 


82  NOTES   ON 


VI.  ELECTRO-MAGNETIC  INDUCTION. 
59.    Induction  Currents  Produced  by  Currents  (§  393). 

Take  two  coils  of  insulated  wire,  wound  so  that  one 
may  be  inserted  inside  the  other,  and  place  a  battery 
in  circuit  with  one  and  a  galvanometer  with  the  other. 
There  is  no  communication  between  the  two  coils,  the 
wire  being  thoroughly  insulated,  but  if  the  coil  in  circuit 
with  the  battery  is  slowly  inserted  in  that  in  circuit  with 
the  galvanometer  (Fig.  147),  the  latter  will  show  a  de- 
flection, which  is  due  to  what  is  known  as  an  induced  cur- 
rent. Experiment  shows  that  this  current  is  not  continu- 
ous, but  exists  only  when  one  coil  is  moved  near  the  other. 
If  the  small  coil  is  inside  the  larger,  no  current  is  ob- 
served in  the  latter  so  long  as  the  current  in  the  former  is 
constant,  but  if  it  is  broken,  an  induced  current  is  observed 
in  the  outer  or  "  secondary  "  coil,  flowing  in  the  same  di- 
rection as  that  formerly  existing  in  the  "primary "  or 
inner  coil.  If  the  circuit  is  closed  in  the  latter  while  it  is 
inside  the  secondary  coil,  the  induced  current  in  the  second- 
ary is  in  the  opposite  direction  to  that  in  the  primary. 
If  the  primary  coil  is  gradually  removed,  a  current  is  in- 
duced in  the  secondary  in  the  same  direction  as  that  in  the 
primary.  There  is  no  current  in  the  secondary,  unless 
there  is  some  change  either  in  the  strength  of  the  primary 
current,  or  in  its  position  relatively  to  the  secondary  coil. 

The  same  effects  are  produced  by  moving  a  magnet  in 
or  near  a  coil  in  circuit  with  a  galvanometer.  If  in  this 
case  the  direction  of  the  Amperian  currents  (see  Note  42) 
be  considered,  the  induced  current  in  the  coil  will  be  in 


ELECTRICITY   AND   MAGNETISM.  03 

the  opposite  or  in  the  same  direction  as  that  of  the  Am- 
perian  currents,  as  the  magnet  is  introduced  or  withdrawn 
from  the  coil.  (See  Fig.  146.) 

Prof.  Thompson  generally  uses  the  word  "  direct"  as 
applied  to  currents,  to  represent  a  positive  current,  one 
flowing  in  the  direction  in  which  the  hands  of  a  watch 
move,  but  in  explaining  these  experiments  he  has  used  it 
in  another  sense  as  meaning  a  current  in  the  same  direc- 
tion, and  confusion  is  apt  to  result.  Whether  a  direct  or 
inverse  current  exists  in  a  secondary  circuit  depends  not 
only  on  the  motion  of  the  primary  coil,  but  on  the  direc- 
tion of  the  current  in  it,  but  his  explanation  does  not  in- 
clude the  latter  at  all.  Reserving  the  terms  direct  and 
inverse  to  apply  to  the  positive  and  negative  directions, 
the  summing  up  on  Page  360  may  be  expressed  thus  : 


By 
means 
of  a 

Momentary    currents 
in  the  opposite  direc- 
tion are  induced  in  the 
secondary  circuit 

Momentary  currents 
in  the  same  direction 
are  induced  in  the  sec- 
ondary circuit 

Magnet 

while  approaching. 

while  receding. 

Current 

while  approaching, 
or  beginning, 
or  increasing. 

while  receding, 
or  ending, 
or  decreasing. 

These  rules,  when  applied  to  magnets,  call  for  a  con- 
ception of  imaginary  currents  flowing  around  them.  By 
considering  the  induction  of  the  current  to  be  due  to  the 
movement  with  or  against  magnetic  forces,  all  the  above 
relations  may  be  expressed  by  the  rule  given  in  §  394  (i.). 


84  NOTES  ON 


"  A  decrease  in  the  number  of  lines  of  force  which  pass 
through  a  circuit  produces  a  current  round  the  circuit 
in  the  positive  direction  (i.  e.,  a  'direct'  current);  while 
an  increase  in  the  number  of  lines  of  force  which  pass 
through  the  circuit  produces  a  current  in  the  negative 
direction  round  the  circuit"  In  the  application  of  this 
rule,  care  must  be  taken  to  look  along  the  lines  of  force  in 
their  positive  direction,  that  in  which  a  north  pole  tends 
to  move,  as  a  current  which  appears  to  be  direct  when 
viewing  it  from  one  side  is  inverse  if  seen  from  the  other, 
and  the  rule  is  worthless  if  misapplied. 

60.  Determination  of  the  Induced  Electromotive  Force 
(§  394,  "•). 

From  the  foregoing  rules,  it  is  seen  that  a  current  is 
induced  in  a  closed  circuit  only  when  there  is  some 
change  in  the  number  of  magnetic  lines  of  force  enclosed 
by  the  circuit.  This  change  may  be  due  either  to  the 
motion  of  the  pole,  to  that  of  the  circuit,  or  to  the  change  of 
current  strength,  if  the  field  is  due  to  a  current.  Each  of 
these  three  cases  requires  the  expenditure  of  energy  in 
some  form,  and  without  such  expenditure  there  is  no 
induced  current.  We  are  therefore  led  to  look  on  the 
energy  of  the  induced  current  as  directly  derivable  from 
the  energy  expended  in  producing  the  change  in  the  field. 

Let  a  current  flow  from  a  battery  through  a  coil  placed 
in  a  magnetic  field.  As  already  shown  (Note  36)  the 
work  done  in  producing  any  displacement  of  the  coil  is 
—  C  (TV,  —  A?i),  or  if  dN  is  the  change  in  the  number  of 
lines  of  force  passing  through  the  circuit  in  the  time  dlt 
this  may  be  written  as  —  C.  dN.  The  current  in  flowing 
through  the  coil  heats  it,  the  amount  of  heat  being  (|  36, - 
CaAJ  .  dt.  If  the  coil  is  placed  so  that  the  lines  offeree  of 
the  field  pass  through  it  in  the  wrong  direction,  it  will 


ELECTRICITY   AND   MAGNETISM.  85 

move,  doing  work  which  is  due  to  the  original  energy  of 
the  current  in  the  circuit.     Hence 

Energy  of  the  current  =  heating  effect  +  work  done  ; 

or,  CE  .  dt  =  C*  7?.  dt  +  C  .  dN 

.-.x=cs  +  ™  ......  o> 

CR  is  the  E.  M.  F.  in  the  coil,  while  E  is  that  originally 
due  to  the  cell.     The  former  is  less  than  the  latter  by 

—  -  ,  which   must  also  be  an  E.  M.  F.   and  due  to   the 
at 

work  done  by  the  coil.     As   a   result,  therefore,  of  the 
motion  of  the  coil  in  the  field,  the  E.  M.  F.    originally  in 

dN 
circuit  is  diminished  by  the  amount  —7-  .    The  E.  M.    F. 

remaining  in  the  circuit  is 

«  =  *-  f  ......  « 


The  induced  E.  M.  F.  is  therefore  measured  by  the 
rate  of  change  in  the  number  of  lines  of  force  which  pass 
through  the  circuit,  and  is  opposite  in  direction  to  that 
originally  existing,  which  caused  the  motion.  By  increas- 
ing this  rate,  by  diminishing  dt,  or,  what  is  the  same  thing, 

dN 
making  the  velocity  of  motion  greater,  -—  may  be  made  to 

equal  or  exceed  E,  and  the  direction  of  the  induced  cur- 
rent would  therefore  be  the  same  whether  there  was  any 
E.  M.  F.  to  be  overcome  or  not.  If  a  battery  current 
flows,  the  induced  current  diminishes  it  ;  if  there  is  no  bat- 
tery current,  the  coil  would  have  to  be  moved  by  external 
agency,  and  the  induced  current  is  in  the  opposite  direc- 


86  NOTES   ON 


tion  to  a  current  which  would  cause  the  motion.     This  is 
seen  directly  from  (2).     If  E  —  O 


The  fact  that  the  induced  current  acts  in  an  opposite 
direction  to  that  causing  the  motion  is  easily  deduced  from 
the  principle  of  conservation  of  energy,  for,  if  a  current 
flowing  in  a  given  direction  caused  motion  of  the  circuit, 
and  this  motion  induced  a  current  in  the  same  direction  as 
the  original,  it  would  increase  the  motion  and  consequently 
the  energy  of  the  system.  Lenz's  law,  given  in  $  396,  is 
therefore  a  direct  result  of  the  conservation  of  energy. 

On  the  supposition  that  the  original  E.  M.  F.  is  zero,  or 
that  there  is  no  current  flowing  when  the  coil  is  at  rest, 

O  =  C*RM  +C.  dN  .....     (4) 

The  original  energy  being  zero,  the  energy  of  the  current 
when  the  coil  is  moved  can  be  obtained  only  from  the 
work  done  in  moving  the  coil,  or 

Work  done  in  moving  coil  =  heating  effect  +  work  do^p 
by  the  coil. 

If  the  coil  does  no  work,  the  total  energy  appears  as  heat 
in  the  circuit. 

dN 

The   E.  M.    F.  induced  in   the   circuit  being  —  the 

dN 
current  is  equal  to          ,  in  which  R  is  the  total  resistance 

in  circuit. 

61.  Practical  Rule  for  Direction  of  Induced  Current 

(§  395). 

From  the  rule  given  in  $  186,  "  Suppose  a  man  swim- 
ming in  the  wire  with  the  current,  and  that  he  turns  so 


ELECTRICITY   AND    MAGNETISM.  87 

as  to  look  along  the  lines  of  force  in  their  positive  di- 
rection, then  he  and  the  conducting  wire  with  him  will 
be  urged  towards  his  left,"  combined  with  Lenz's  Law 
(Note  62),  the  following  rule  for  the  direction  of  the  in- 
duced current  in  a  conductor  is  easily  deduced.  Suppose 
.a  man  swimming  in  any  conductor  to  turn  so  as  to  look 
along  the  lines  of  force  in  their  positive  direction  ;  then 
if  he  and  the  conductor  be  moved  toward  his  left  hand 
he  will  be  swimming  against  the  current  induced  by 
this  motion  ;  if  he  be  moved  toward  his  right  hand  the 
current  will  be  with  him.  Through  some  error  this  rule 
is  given  incorrectly  in  $  395,  and  differs  there  from  the 
rule  as  given  by  Prof.  Thompson  in  his  Cantor  lectures. 

62.  Lenz's  Law  (§  396). 

In  Note  60  it  was  shown  that  in  accordance  with  the 
principle  of  the  conservation  of  energy,  the  induced  cur- 
rent resulting  from  any  motion  of  a  conductor  must  be  in  the 
opposite  direction  to  that  of  the  current  which  would  cause 
the  motion;  Lenz  deduced  this  relation  independently, 
and  his  statement  that  "  in  all  cases  of  electromagnetic 
induction  the  induced  currents  have  such  a  direction 
that  their  reaction  tends  to  stop  the  motion  which  pro- 
duces them  "  is  known  as  Lenz's  Law.  As  an  illustration 
of  the  use  of  the  law,  suppose  a  magnet  to  be  inserted  in  a 
hollow  coil.  The  induced  currents  must  be  in  such  a 
direction  as  to  oppose  the  motion.  As  opposite  currents 
repel  each  other,  the  current  induced  in  the  coil  will  be 
opposite  in  direction  to  the  Amperian  current  of  the 
magnet.  If  the  magnet  is  withdrawn,  the  withdrawal 
would  be  opposed  by  a  current  in  the  same  direction  as 
the  Amperian  current,  and  by  the  law  a  current  would 
therefore  be  induced  in  that  direction.  The  same  reason- 
ing applies  to  currents. 


88  NOTES  ON 


63.  Mutual  Induction  of  Two  Circuits  (§  397). 

In  \  320  and  Note  39  it  was  seen  that  two  circuits 
tended  to  place  themselves  in  such  a  position  as  to  inclose 
as  many  of  each  other's  lines  of  force  as  possible,  and  the 
number  inclosed  when  each  carried  unit  current  was 
denoted  by  M.  It  has  since  been  shown  that  any  move- 
ment of  either  circuit  induces  a  current  in  the  other,  and 
consequently  changes  the  value  of  M.  This  quantity 
is  therefore  called  the  "  coefficient  of  mutual  induction." 
By  a  course  of  reasoning  similar  to  that  in  Note  18  it  may 
be  shown  that  the  force  just  outside  the  plane  of  a  voltaic 
circuit  of  unit  area  is4?rC,  and  if  6"  is  the  area  inclosed  by 
the  circuit  the  total  force  is  ^nCS,  which  becomes  ^nS 
when  C  is  of  unit  strength  and  tytnS  when  there  are  n 
turns  each  of  area  S.  But  as  the  number  of  lines  of  force 
is  numerically  equal  to  the  strength  of  the  field,  this  num- 
ber would  therefore  be  4?r5,  and  if  all  these  lines  passed 
through  the  other  circuit,  the  maximum  value  of  Mis  $7tS 
when  the  two  circuits  are  coincident. 

64.  Self-induction  (§  404). 

The  extra  current  is  a  current  induced  in  the  same 
conductor  in  which  the  original  current  flows.  By  Lenz's 
law,  or  by  the  table  in  Note  59,  when  a  current  begins  in  a 
conductor,  a  momentary  current  is  induced  in  the  opposite 
direction,  and  this  phenomenon  is  noticeable  as  well  in 
the  original  circuit  as  in  another  near  it.  The  current 
in  beginning  is,  therefore,  opposed  by  an  induced  current 
in  the  opposite  direction,  and  its  increase  is  made  more 
gradual,  and  more  time  is  necessary  for  it  to  gain  its  full 
strength.  The  fact  that  the  primary  current  is  greatly 
reduced  by  the  induced  current  accounts  for  the  fact  that 
when  a  circuit  is  closed,  but  little  of  a  spark  is  seen.  After 


ELECTRICITY  AND   MAGNETISM.  89 

the  current  attains  its  normal  strength  it  remains  un- 
affected by  induction,  unless  acted  upon  by  external  causes  ; 
but  if  the  circuit  is  broken,  the  law  and  table  already  re- 
ferred to  indicate  a  current  in  the  same  direction  as  the 
primary,  retarding  its  decrease.  As  this  current  prac- 
tically reinforces  the  primary,  the  spark  at  breaking  the 
circuit  is  much  brighter  than  at  making. 

The  extra  current  being  strictly  an  induced  current  is 
subject  to  the  general  laws  of  induction.     The  induced  E, 

dN 

M.  F.  is  therefore -jr.     If  dN  is  constant,  as  it  is  with 

at 

any  given  current,  the  E.  M.  F.  of  the  extra  current  varies 
inversely  as  dt,  or  the  more  quickly  the  circuit  is  closed 
or  broken  the  greater  the  extra  current.  The  value  of  L 
in  §  404  determines  the  other  important  relation,  that  the 
self-induction  varies  as  the  square  of  the  number  of  turns, 
and  as  the  resistance  of  the  coil  increases  only  as  the 
number  of  turns,  the  extra  current  is  much  greater  the 
more  turns  the  coil  possesses.  This  is  of  great  importance 
in  enabling  extra  currents  of  high  E.  M.  F.  to  be  obtained 
when  wished,  or  avoided  when  they  would  be  detrimental. 

65.  Helmholtz's  Equations  (§  405), 

The  current  is  prevented  by  its  self-induction  from  obtaining 
its  full  strength  immediately.     The  electromotive  force  of  the 

induced  current  is  —  L  .  —  >     L  being  the   coefficient  of   self- 
dt 

induction,  and  the  current  acting  in  opposition  to  the  primary 

T     sJ  r* 

current  is  —  •  —  .    In  any  interval  of  time,  *//,  after  the  circuit  is 
R    at 

closed,  the  current  has  a  strength  of 

E        L    dC  dC  R     . 


90  NOTES   ON 


and  the  current  at  a  time  /  from  the   instant  of  closing  the  cir- 
cuit is 


66.  Induction  Coil  (§  398). 

It  is  important  to  remember  that  in  the  induction  coil 
there  are  two  circuits,  not  only  independent  of  each  other 
but  carefully  insulated.  The  primary  coil,  or  the  one  im- 
mediately surrounding  the  iron  core,  is  of  comparatively 
few  turns,  and  of  low  resistance,  that  a  given  E.  M.  F.  may 
cause  as  powerful  a  current  as  possible  and  consequently 
induce  as  many  lines  of  force  as  possible  through  the  core. 
If  the  primary  current  is  made  and  broken  rapidly,  this 
number  of  lines  is  alternately  added  and  subtracted  at 
very  short  intervals,  and  the  E.  M.  F.  induced  in  the  second- 
ary coil,  through  the  axis  of  which  all  the  lines  of  the 
primary  pass,  is  therefore  very  great.  From  the  general 

dN 
formula  E  = j-  >  the  induced  E.  M.  F.  in  the  secondary 

is  evidently  increased  by  using  greater  battery  power  in 
the  primary  or  by  making  and  breaking  the  primary  cir- 
cuit with  greater  rapidity.  The  E.  M.  F.  of  the  secondary 
circuit  becomes  so  great  that  extreme  care  has  to  be  taken 
with  the  insulation,  and  parts  of  the  coil  at  widely  dif- 
ferent potentials  must  not  be  brought  near  together. 
It  is  noticeable  that  the  quantity  of  electricity  passing 


ELECTRICITY   AND    MAGNETISM. 


in  the  secondary  coil  is  extremely  small.  This  is  at  once 
apparent  when  it  is  considered  that  the  energy  of  the 
secondary  coil,  which  may  be  expressed  as  C'E',  is  all 
derivable  from  that  of  the  primary  coil  CE  and  cannot 
exceed  it.  If  the  energy  in  the  two  coils  is  assumed  equal, 

C  :   C  :  :  E'  :  E, 

and  the  enormous  increase  of  E.  M.  F.  in  the  secondary  is, 
therefore,  attended  with  a  great  reduction  in  the  quantity 
of  electricity  passing.  As  all  the  energy  of  the  primary 
current  cannot  be  transferred  to  the  secondary,  this  pro- 
portion is  not  strictly  correct,  but  it  illustrates  the  im- 
portant point  that  an  induction  coil  which  might  kill  a 
man  could  not  heat  a  wire  red  hot,  or  perform  other  work 
where  quantity  was  necessary. 

The  condenser  is  made  of  sheets  of  tin  foil  insulated 
from  each  other  by  paper  soaked  in  paraffine.  Alternate 
sheets  are  connected  throughout,  so  as  to  form  two  large 
coatings.  The  action  may  be  understood  from  the  figure. 
The  current  passes  from  the  battery  through  Wto  I  the 


Fig.  23. 

interrupter,  and  thence  through  o  back  to  the  battery. 
The  core,  on  being  magnetized,  attracts  /,  breaking 
the  current  at  o.  If  there  were  no  condenser  the  extra 
current  would  leap  across  from  /  to  o  in  a  bright  spark, 
but  when  the  condenser  is  used  it  darts  into  it,  charging  P 


92  NOTES   ON 


positively  and  N  negatively,  but  immediately  afterwards 
the  two  charges  re-combine,  the  positive  charge  passing 
from  P  to  L  WBNt  demagnetizing  the  core  and  making 
the  "  break  "  more  rapid,  and  also  opposing  the  current 
at  the  "  make."  In  this  way  the  time  dt  of  the  break  is 
diminished  while  that  of  the  make  is  increased,  and  the 
E.  M.  F.  induced  in  the  secondary  coil  at  the  former  is, 
therefore,  much  the  greater.  By  separating  the  poles  of 
the  secondary  circuit,  they  may  be  placed  so  that  the 
break  or  similar  current  can  strike  across  them  while  the 
make  or  reverse  current  cannot. 


ELECTRICITY   AND   MAGNETISM. 


93 


VII.  DYNAMO  MACHINES. 
67.  General  Principle  of  Dynamo  Machines  (§  407). 

The  principle  underlying  all  production  of  electricity  by 
machines,  is  that  of  Note  59,  that  if  a  coil  of  wire  is  moved 
in  a  magnetic  field  a  current  is  induced  in  the  coil.  The 
successive  machines  have  simply  been  developments  of 
this  fact,  improvements  having  been  made  either  in  the 
distribution  of  the  lines  of  force  in  the  field,  or  in  the  con- 
struction and  movement  of  the  coils.  In  the  first  machines, 
those  of  Pixii,  Saxton  and  Clarke,  permanent  steel  magnets 
were  used,  but  only  a  portion  of  the  lines  due  to  these 
poles  were  cut  by  the  coils,  and 
the  machines  were,  therefore,  in- 
efficient. As  a  rule,  the  coils 
moved  in  front  of  the  poles,  in- 
tercepting the  lines  passing  off 
from  the  poles  in  one  direction 
only.  The  same  general  prin- 
ciple was  followed  in  the  Holmes 
and  Alliance  machines,  there 
being  a  greater  number  of  mag- 
nets and  coils,  with  a  poor  dispo- 
sition of  the  different  parts.  The 
introduction  of  Siemens'  armature  Flg'  24  A' 

in  1857,  was  a  great  step  in  advance.  N  and  6*  are  the 
poles  of  several  horseshoe  magnets  bolted  together  side  by 
side,  and  between  the  opposite  poles  rotates  a  soft  iron  cyl- 
inder Con  which  the  wire  is  coiled.  This  armature  is  thus 


— N— 


W 


94  NOTES  ON 


placed  in  the  strongest  part  of  the  field,  the  greater  number 
of  lines  of  force  passing  directly  from  N  to  S  through  the 
core  C,  and  the  construction  admits  of  the  coils  approach- 
ing the  poles  very  closely.  Very  strong  currents  were  ob- 
tained from  machines  in  which  this  armature  was  used, 
but  great  difficulty  was  experienced  from  the  heating  of 
the  armature.  If  a  disk  of  any  metal  is  rotated  rapidly 
between  the  poles  of  a  powerful 
magnet  it  becomes  greatly  heated 
by  the  currents  which  are  induced 
in  the  metal.  Tyndall  melted 
fusible  metal  in  a  copper  tube 
Fig.  24  B.  rotated  rapidly  in  a  strong  field. 

The  induced  currents  are  in  this  case  in  the  metal  and 
not  in  the  coil  and  are  generally  known  as  "  Foucault " 
currents.  The  heat  evolved  depends  on  the  form  and 
material  of  the  armature  and  on  its  velocity,  and  heat  from 
this  cause  has  always  been  a  serious  objection  to  the  early 
form  of  Siemens'  armature.  The  next  step  \vas  to  have 
two  armatures  in  the  same  machine  ;  one  rotating  between 
the  poles  of  permanent  magnets  inducing  a  current  which 
passed  through  the  coils  of  a  large  electromagnet,  between 
the  poles  of  which  the  other  armature  was  placed.  The 
adoption  of  electro-magnets  greatly  intensified  the  field, 
and  as  the  current  causing  them  was  also  generated  by 
the  machine,  a  great  gain  of  power  and  efficiency  was 
secured.  The  machines  of  Ladd  and  Wilde  were  of  this 
type.  The  above  were  called  magneto-electric  machines, 
as  they  all  depended  on  permanent  magnets  to  start  them, 
but  in  1867  the  permanent  magnets  were  suppressed,  the 
current  from  the  armature  passing  through  the  coils  of  the 
electro-magnet,  the  "  field  "  coils  being  in  series  with  the 
external  circuit  and  armature.  When  the  machine  was 
stopped  it  was  found  that  the  cores  of  the  electro-magnets 


ELECTRICITY  AND   MAGNETISM.  95 

possessed  sufficient  residual  magnetism  to  induce  a  cur- 
rent in  the  armature  when  it  was  started  again,  and  this 
current  once  induced,  strengthened  the  electro-magnets 
and  in  turn  induced  more  current.  Machines  of  this  type 
were  called  dynamo-electric  or  simply  dynamo  machines. 
The  advantages  they  possess  over  the  magneto-electric  are 
greater  power,  the  field  being  stronger ;  and  greater  econ- 
omy, the  magnets  being  of  wrought  iron  instead  of  steel. 
The  distinction  between  magneto  and  dynamo  machines 
is  now  so  slight,  by  the  introduction  of  a  variety  of  new 
types,  that  it  is  hardly  worth  preserving.  As  has  already 
been  shown,  the  energy  of  the  induced  current  is  deriva- 
ble from  the  energy  expended  in  moving  the  coil,  and  Prof. 
Thompson  in  his  Cantor  lectures  has  given  a  broad  defi- 
nition of  a  dynamo  machine  as  "a  machine  for  converting 
energy  in  the  form  of  dynamical  power  into  energy  in  the 
form  of  electric  currents  by  the  operation  of  setting  con- 
ductors (usually  in  the  form  of  coils  of  copper  wire)  to 
rotate  in  a  magnetic  field."  Accepting  this  definition,  the 
theory  of  the  dynamo  is  best  understood  by  a  reference 
to  the  laws  of  electro-magnetic  induction  already  examined. 
The  induced  electromotive  force  in  a  conductor  moving  in 

dN 

a  magnetic  field  is  E  — — .  As  an  illustration,  exam- 
ine the  case  of  a  coil  spinning  in  a  uniform  field,  and  the 
application  of  the  formula  to  dynamos  may  be  considered 
later.  Suppose  a  coil,  as  in  Figure  25,  rotating  on  a  vertical 
axis,  the  lines  of  force  passing  from  the  reader  down 
Through  the  paper  perpendicularly.  It  incloses  a  maxi- 
mum number  of  lines  offeree,  and  if  rotated  so  that  the 
right-hand  edge  comes  to  the  front,  while  the  left-hand 
goes  behind  the  paper  it  will  inclose  a  constantly  decreas- 
ing number  of  lines,  and  a  positive  current  will  be  in- 
duced. The  E.  M.  F.  will  at  first  be  small,  as  the  rate  of 


96  NOTES  ON 


change  is  small,  the  edges  of  the  coil  moving  almost  along 
the  lines  of  force.     The  rate  will  gradually  increase  until 
the  coil  has  moved  through  one  quadrant  and  is  edge  on 
to  the  observer,  when,  as  the  motion 
of  the  edges  is  at  right-angles  to  the 
lines,  the  rate,  and  consequently  the 
E.  M.  F.,  is  a   maximum.     In  this 
position  the  coil  incloses  no  lines  of 
force,  and  during  the  second  quad- 
rant  it  will  move,  inclosing  an  in- 
creasing   number,     and     inducing, 
therefore,  an   inverse  current.     But 
the  side  of  the  coil  now  seen  is  the 
Fig.  25.  opposite  to  that  in  view  during  the 

first  quadrant,  and  the  inverse  current  is,  therefore,  in  the 
same  absolute  direction  in  the  coil  as  the  former  direct 
current.  During  the  second  quadrant  the  rate  and  E.  M. 
F.  decrease,  becoming  a  minimum  when  the  coil  has  com- 
pleted a  half  revolution  and  is  again  in  the  plane  of  the 
paper.  On  entering  the  third  quadrant,  the  number  of 
lines  inclosed  decreases,  and  a  direct  current  is  induced  ; 
but  as  the  same  side  of  the  coil  is  presented  to  the  observer 
as  in  the  second,  the  direction  of  the  current  is  reversed 
in  the  coil.  In  the  fourth  quadrant  the  number  of  inclosed 
lines  increases,  but  the  other  side  of  the  coil  is  toward  the 
observer,  so  that  the  absolute  direction  of  the  current  is 
the  same  as  in  the  third.  The  general  direction  of  the 
current  is,  therefore,  downward  in  that  part  of  the  coil  in 
front  of  the  paper,  and  upward  in  the  other  half ;  but  as 
regards  the  coil  itself,  the  direction  of  the  current  changes 
twice  in  every  revolution,  the  point  of  change  being 
where  the  circuit  incloses  the  maximum  number  of  lines 
of  force.  By  the  use  of  a  commutator  which  shifts  its 
connections  at  this  point  of  the  revolution,  the  current 


ELECTRICITY   AND   MAGNETISM.  97 

may  be  made  to  flow  in  one  direction  in  the  exterior 
circuit. 

Considering  this  coil  as  the  armature  of  a  dynamo  ma- 
chine, it  is  apparent  that  the  current  could  be  kept  in  one 
direction  in  the  exterior  circuit,  but  would  be  of  varying 
strength.  If  another  coil  were  fixed  on  the  same  axis  but 
at  right  angles  to  the  first,  its  E.  M.  F.  would  be  a  max- 
imum when  that  of  the  first  was  a  minimum,  making 
the  current  in  the  external  circuit  more  nearly  uniform. 
By  increasing  the  number  of  coils  a  practically  uniform 
current  could  be  obtained,  but  at  the  expense  of  a  very 
complicated  commutator. 

68.  Electromotive  Force. 

dN 

From  the  formula  E  = —  it  is  evident,  I.  That  the 

at 

E.  M.  F.  varies  as  the  rate  of  change  of  the  field.  For  a 
constant  time  dt,  the  rate  varies  as  the  number  of  lines 
taken  out  or  introduced,  and  the  field  should  therefore 
be  intense.  Mere  intensity  is  not,  however,  enough, 
as  a  coil  could  be  moved  in  the  most  intense  uniform 
field  without  inducing  any  current.  The  field  must  be  so 
arranged  that  the  coil  either  passes  from  a  maximum 
positive  to  maximum  negative,  or  what  amounts  to  the 
same  thing,  that  it  rotates  in  a  constant  field,  the  lines 
being  alternately  added  and  subtracted.  In  this  case  the 
number  of  lines  should  be  a  maximum  or  the  intensity  cf 
the  field  as  great  as  possible. 

Perfect  working  in  a  dynamo  requires  a  constant 
change  of  E.  M.  F.,  and  consequently  a  constant  rate.  If 
a  large  coil  revolves  between  the  poles  of  two  bar  mag- 
nets, and  no  iron  is  present  to  modify  the  distribution  of 
the  lines  of  force  in  the  field,  the  greater  part  pass  direct" 
7 


NOTES   ON 


between  the  poles,  and  are  cut  during  a  small  part  of  the 
revolution  of  the  coil,  during  which  time  the  rate  and  in- 
duced E.  M.  F.  are  high,  but  in  other  parts  of  the  revolu- 
tion the  rate  is  very  small.  The  available  E.  M.  F.  is 
induced  suddenly,  but  the  sudden  creation  of  a  current 
causes  high  self-induction  and  temporary  strong  extra 
currents,  which  in  a  dynamo  are  not  only  prejudicial  but 
dangerous,  on  account  of  the  high  E.  M.  F.  they  may 
have.  Idle  wire  in  the  armature  also  reduces  the  current. 
It  is  therefore  desirable  to  prevent  a  concentration  of  the 
lines  of  force  in  a  small  part  of  the  field.  In  a  coil  rotat- 
ing in  a  uniform  field,  the  advantage  of  the  constant  rate 
is  attained  by  the  change  of  the  number  of  lines  inclosed 
in  the  ratio  of  the  sine  of  the  angle  between  the  plane  of 
the  coil  and  the  direction  of  the  lines,  and  as  a  field  tends 
to  become  uniform  would  this  advantage  be  gained. 
To  secure  uniformity  and  prevent  concentration  large 

pieces  of  iron  called 
pole  pieces  are  fre- 
quently attached  to 
the  poles,  partially 
encircling  the  arma- 
ture. By  using  long 
magnets  and  heavy 
pole  pieces,  the  field 
may  be  made  nearly 
uniform.  An  impor- 
tant modification  of 
the  field  arises  from 
the  lines  of  force  due 
to  the  current  in  the 
armature.  Thus  in 
Fig.  26,  representing  a  cross  section  of  a  Siemens  arma- 
ture, A  being  the  end  of  the  commutator  and  TT  the 


Fig.  26. 


ELECTRICITY  AND   MAGNETISM.  99 

commutator  brushes  ;  the  lines  of  force  of  the  field  ordi- 
narily pass  from  N  to  S  in  approximately  straight  lines. 
When  the  armature  is  in  revolution,  each  coil  in  succession 
has  its  maximum  current  when  it  is  in  the  position  C  in 
the  figure,  and  the  effect  of  the  armature  current  is  there- 
fore to  induce  two  poles  in  the  rim  of  the  armature  at  A ' 
and  S'.  The  poles  A7  and  N'  may  be  supposed  to  form  a 
resultant  pole  at  N"  and  5  and  S1  at  S",  and  the  gen- 
eral direction  of  the  lines  of  force  of  the  field  is  therefore 
N"  S".  As  shown  in  the  discussion  of  the  revolving  coil, 
the  commutator  brushes  should  make  contact  at  the 
neutral  points,  at  right  angles  to  the  lines  of  force.  If  the 
lines  of  force  of  the  machine  in  motion  were  in  the  same 
position  as  when  at  rest  the  brushes  might  remain  at 
TT  in  a  line  perpendicular  to  the  lines  of  force  NS.  If 
left  in  this  position,  however,  it  will  be  observed  that 
there  is  a  constant  succession  of  sparks  at  the  brushes, 
which  evidently  do  not  press  at  the  neutral  points.  This 
sparking  may  generally  be  suppressed  by  rotating  the 
brushes  into  positions  T'T'.  The  explanation  is  simple  : 
they  have  been  brought  into  a  line  PP  perpendicular  to 
the  changed  direction  N" S"  of  the  lines  of  force  of  the 
field.  The  stronger  the  current,  the  stronger  the  induced 
pole  of  the  armature  N' ',  and  the  nearer  TV"  is  the  resultant 
pole  N".  The  stronger  the  current,  therefore,  the  more 
the  brushes  must  be  advanced. 

The  different  types  of  dynamos  vary  principally  in  the 
way  in  which  the  field  is  formed.  The  principal  methods 
are: 

(i.)  The  magneto  in  which  the  field  is  due  to  permanent 
magnets.  (2.)  The  separately  excited  dynamo,  a  separate 
machine  being  used,  the  current  of  which  passes  through 
the  field  magnet  coils  of  the  generator.  This  possesses 
the  great  advantage  of  having  a  constant  field,  and  when 


100  NOTES  ON 


several  machines  are  used  in  one  place,  one  may  be 
used  to  actuate  the  field  magnets  of  all  the  others.  (3.) 
The  series  dynamo,  the  field  coils  being  in  the  main  cir- 
cuit. As  the  whole  current  of  the  machine  passes  around 
the  magnets,  an  intense  field  is  produced,  but  with  the 
great  disadvantage  that  any  increase  of  resistance  in  the 
external  circuit  weakens  the  field,  and  consequently  the 
E.  M.  F.,  just  when  a  high  E.  M.  F.  is  necessary  to  over- 
come the  increased  resistance.  (4.)  The  shunt  dynamo 
has  the  field  magnet  coils  in  a  shunt  of  the  main  circuit. 
In  this  type  an  increased  external  resistance  sends  a 
greater  current  through  the  magnet  coils,  causing  a  more 
intense  field  and  a  higher  E.  M.  F.  By  having  the  resist- 
ance in  the  magnet  coils  adjustable,  a  shunt  dynamo  may 
be  made  to  give  a  practically  constant  E.  M.  F.,  whatever 
the  external  resistance  may  be.  (5.)  A  mixture  of  the  last 
two  types  has  been  used  for  special  purposes,  and  is 
known  as  a  series  and  shunt,  or  compound  dynamo.  It  is 
a  shunt  dynamo  having  in  addition  a  number  of  coils  of 
wire  on  its  field  magnets  in  the  main  circuit.  In  a  shunt 
dynamo,  if  the  external  resistance  is  suddenly  increased, 
a  greater  part  of  the  current  flows  around  the  field  coils, 
inducing  a  higher  E.  M.  F.  To  keep  this  constant  the 
speed  would  have  to  be  decreased.  In  a  series  dynamo, 
however,  an  increase  of  external  resistance  diminishes  the 
E.  M.  F.,  and  to  keep  it  constant  the  speed  must  be  in- 
creased. By  making  the  magnet  coils  partly  in  series  and 
partly  in  a  shunt  circuit,  it  is,  therefore,  possible  to  keep 
the  E.  M.  F.  practically  constant  at  a  certain  speed  within 
wide  changes  of  the  external  resistance.  (6.)  Some 
machines  have  two  coils  on  the  armature,  one  of  which 
sends  a  current  through  the  field  coils,  while  the  other  is 
in  the  external  circuit.  (7.)  Alternate  current  machines 
are  machines  giving  currents  first  in  one  direction  and 


ELECTRICITY  AND   MAGNETISM.  1OI 

then  in  the  opposite.  Their  field  is  caused  generally  by 
another  machine,  but  some  types  send  a  part  of  their  own 
current,  rectified  by  a  commutator,  through  the  field  coils. 

2.  The  E.  M.  F.  varies  as  the  velocity. 

This  relation  is  almost  absolutely  true  in  magneto 
machines  and  in  others  having  a  constant  field,  but 
in  the  series  dynamo,  where  the  field  is  itself  a  function  of 
the  current,  the  rate  of  increase  of  E.  M.  F.  is  much 
greater  than  that  of  increase  of  velocity  up  to  the  point  of 
saturation  of  the  magnets,  beyond  which,  the  field  being 
constant,  the  above  relation  holds.  Its  correctness  is  as- 
sumed in  practice. 

A  great  waste  sometimes  occurs  from  the  commutator 
brushes  not  being  adjustable.  As  already  shown,  the 
lines  of  force  of  the  field  are  distorted  by  the  current,  and 
this  distortion  is  greater  as  the  velocity  is  increased.  The 
brushes  must,  therefore,  be  advanced,  or  they  will  take  off 
the  current  at  the  wrong  time,  involving  a  waste  of 
energy  and  causing  "sparking,"  injuring  both  commutator 
and  brushes. 

3.  The  E.  M.  F.  varies  as  the  number  of  turns  of  wire 
in  the  armature. 

dN 
The  formula  E  — -=•   is  derived  from  a  conception 

of  the  work  done  in  moving  a  single  coil  in  a  magnetic 
field.  If  n  coils  were  moved  either  one  by  one  or  all  to- 
gether, n  times  the  work  would  be  done,  and  n  times  the 
E.  M.  F.  induced.  Increasing  the  number  of  turns  in- 
creases the  internal  resistance  in  the  same  ratio,  but  if  the 
external  resistance  is  large  there  is  a  gain  by  taking  more 
turns  of  wire  on  the  armature. 

4.  The  E.  M.  F.  is  greatest  when  the  coil  cuts  the  lines 


102  NOTES  ON 


of  force  at  right  angles.  The  rate  of  change  is  then 
greatest,  and  hence  the  electromotive  force. 

5.  The  E.  M.  F.  varies  as  the  area  of  the  coil.  If  the 
field  were  uniform  the  gain  would  be  directly  as  the  area. 
In  any  case  it  varies  directly  as  the  number  of  lines  of 
force  inclosed,  and  there  is,  therefore,  generally  speaking, 
an  advantage  in  having  the  coil  of  large  area. 

In  the  five  considerations  on  which  the  E.  M.  F.  of  in- 
duction depends,  the  only  variable,  after  the  machine  is 
made,  is  the  velocity.  Several  of  the  types  referred  to 
admit  of  a  partial  adjustment  to  meet  changed  circum- 
stances, but  in  general  a  machine  should  be  adapted  to 
the  work  expected  of  it,  and  should  not  be  expected  to  be 
efficient  under  very  different  conditions.  Although  the 
velocity  may  be  easily  varied,  it  cannot  be  indefinitely  in- 
creased without  mechanical  injury. 

69.  Efficiency. 

A  dynamo  has  properly  two  efficiencies.  As  it  is  a 
vehicle  for  the  transformation  of  mechanical  into  electrical 
energy  its  gross  efficiency  is  the  ratio  of  the  current 
energy  to  the  mechanical  energy  actually  applied  to  the 
machine.  If  an  engine  developing  16  H.  P.,  is  working 
a  dynamo,  two  H.  P.  being  lost  in  transmission  to  the 
dynamo,  in  friction  and  in  overcoming  the  inertia  of  the 
engine,  only  14  H.  P.  are  actually  applied  to  turn  the 
armature.  If  in  this  case  the  electrical  energy  developed 
by  the  dynamo,  was  10  H.  P.  the  gross  efficiency  is  \\t 
or  71  per  cent.  A  good  dynamo  possesses  an  efficiency 
of  from  90  to  95  per  cent,  when  working  under  the 
most  favorable  conditions,  and  therefore  far  surpasses 
any  other  machine  in  its  capacity  for  transforming 
energy. 


ELECTRICITY   AND    MAGNETISM.  103 

The  ordinary  use  of  the  dynamo  is  to  produce  light. 
Whatever  its  use  may  be,  all  the  electrical  energy  not 
utilized  in  producing  the  desired  result  is  practically 
wasted.  The  net  efficiency  is  the  ratio  of  the  electrical 
energy  in  the  external  circuit  to  the  mechanical  energy 
applied  to  the  armature.  If  in  the  above  case  only  6 
H.  P.  existed  in  the  external  circuit,  the  net  efficiency 
=  -,&4-,  or  44  per  cent.  The  distribution  of  the  energy  in 
the  circuit  is  one  of  the  most  important  problems  relating 
to  dynamos.  The  total  work  in  circuit  is,  by  Note  55, 
C^Rt,  R  being  the  total  resistance,  consisting  of  internal  r, 
and  external  /.  The  work  is  then  CV  +  CY,  and  the 
ratio  of  the  work  done  in  the  machine  to  that  in  the  ex- 

C'r        r 
ternal   circuit  is  -—  =  —  •       The  fraction  of  the  total 

C-l 
electrical  energy  in  +he  external  circuit  is  similarly  -^—5 

=  — •     That  this  proportion  should  be  great,  /  must  be 

nearly  equal  to  R,  or  in  other  words,  the  resistance  of  the 
machine  must  be  small  compared  with  that  of  the  circuit. 
From  the  above  the  important  relation  is  evident,  that 
the  distribution  of  energy  in  an  electrical  circuit  is 
determined  by  the  distribution  of  the  resistances  in  cir- 
cuit. 

The  efficiencies  may  now  be  calculated.  Let  a  —  resist- 
ance of  armature, /that  of  the  field  coils,  /of  the  external 
circuit,  and  R  be  the  total  resistance.  E  is  the  electro- 
motive force,  and  Cthe  current  in  circuit. 

The  gross  efficiency  of  a  series  dynamo 

Work  of  current 
=     H.  P.  applied  ' 


104  NOTES   ON 


This  by  Note  55  is 
CV? 


_  _ 

H.  P.  applied  ~  746  x  H.  P.  applied* 

This  energy  is  given  off  in  all  parts  of  the  circuit,  that 
in  the  armature  being  C*a  (in  watts),  that  in  the  field 
coils  Cy,  and  in  the  external  circuit  C2/. 

The  net  efficiency  is 

Work  in  external  circuit- 

H.  P.  applied 
and  this  is 

CW 

"746  cn 


H.  P.  applied        746  x  H.  P.  applied 
The  energy  wasted  as  heat  in  the  machine  is  C2  (a  +  f) 
and  the  ratio  of  energy  wasted  is /      /    That  this 

may  be  small  the  internal  resistance  (a  +  f)  must  be 
small  in  comparison  with  the  external.  The  energy 
wasted  takes  the  form  of  heat,  and  is  thus  not  only 
wasted  but  directly  harmful,  as  heating  of  the  machine 
increases  its  resistance  and  thereby  increases  the  ratio  of 
wasted  energy. 

In  the  shunt  dynamo  the  relations  are  more  complex, 
as  the  current  in  the  various  branches  of  the  circuit  is 
different. 

Let  a  =  armature  resistance  and  A  armature  current, 
f  =  field  coil  "  "    .F  current  in  field  coil, 

/  =  external  "  "    L      "        "    external 

circuit, 

R  =  Total  resistance  =  a   +  -, — —s 


ELECTRICITY  AND   MAGNETISM.  105 

Work  in  armature  =  A-a,  in  field  coils  F*f,  and  in  ex- 
ternal circuit  Ul. 

Total  electrical  energy  =  A*R  =  A1  (a  +       ^     J . 
Gross   efficiency  = 


H.  P.  applied  746  x  H.  P.  applied 

To  find  the  net  efficiency,  the  ratio  of  the  electrical  en- 
ergy utilized  is 

'"-—         r. 


( 


and  by  multiplying  this  into  the  value  of  the  gross  efficiency, 
previously  obtained,  the  product  is  the  net  efficiency. 

The  above  expression  contains  only  resistances.  If  L  is 
measured  the  net  efficiency  is  evidently 

Z,V 
746  x  H.  P.  applied 

70.  Electromotive  Force  in  Circuit. 

In  using  CE  to  calculate  the  electrical  energy  in  any 
case,  E  must  not  be  taken  as  the  difference  of  potential  at 
the  machine  terminals.  Calling  this  difference  of  potential 
E',  and  considering  the  case  of  the  series  dynamo  as  being 
more  simple,  we  have  from  Kirchhoff  's  second  law,  E'  =  Cl, 
I  being  the  external  resistance, 

or        c  =  ?- ,  but  from  Ohm's  Law  C  =  j^— 


106  NOTES   ON 


Whenever  E'  is  measured  E  must  be  calculated  if  the 
total  electrical  energy  is  to  be  computed.  The  product 
CE'  is  evidently  the  energy  in  the  external  circuit,  and  is 
less  than  the  total  energy  by  the  quantity  CV  expended 
in  the  machine.  The  total  energy  is,  therefore,  CE'  + 
C*r  =  C-(l  -f  r).  If  the  resistances  are  all  known  the 
total  energy  may  be  calculated  from  the  last  formula  with- 
out any  risk  of  error. 

It  has  been  shown  by  Sir  William  Thomson  that  in 
shunt  dynamos  the  best  results  are  obtained  when  the 
external  resistance  is  a  mean  proportional  between  the 
resistance  of  the  magnet  coils  and  that  of  the  armature, 
the  latter  being  small  in  comparison  with  the  resistance 
of  the  magnet  coils. 

71.  Siemens'  Machine  (§  409,  Fig.  151). 

This  is  a  shunt  dynamo.  The  armature  is  similar  in 
shape  to  Siemens'  armature  already  described,  being  a 
cylindrical  drum,  but  having  several  coils  coiled  on  it 
lengthwise  instead  of  one.  There  are  as  many  divisions 
of  the  commutator  as  there  are  coils,  the  divisions  being 
longitudinal.  An  eight-coil  machine  has,  therefore,  its 
commutator  ring  divided  into  eight  segments,  to  each  of 
which  connect  the  ends  of  two  coils.  The  other  ends  of 
these  coils  are  connected  to  other  commutator  divisions, 
so  that  the  eight  coils  are  all  in  a  continuous  circuit  be- 
tween the  commutator  brushes,  so  wound  that  in  all  eight 
the  current  at  any  given  instant  flows  in  the  same  direc- 
tion. In  some  coils  the  E.  M.  F.  is  greater  than  in  others, 
but  as  there  are  so  many,  the  total  E.  M.  F.  of  all  in  series 
varies  but  slightly  from  time  to  time,  and  the  current  is, 
therefore,  practically  constant.  By  placing  the  commu- 
tator brushes  opposite  each  other,  they  are  in  contact  with 
points  of  the  circuit  differing  most  widely  in  potential,  and 


ELECTRICITY   AND    MAGNETISM.  107 

a  permanent  difference  of  potential  is  therefore  maintained 
between  the  terminals  of  the  machine.  The  induction  of 
the  current  in  any  one  coil  is  analogous  to  that  in  the  coil 
described  in  Note  67. 

72.  The  Gramme  Machine  (§  410,  Fig.  153). 

The  Gramme  is  generally  a  series  dynamo,  although 
sometimes  separately  excited,  and  sometimes  having  its  field 
coils  excited  by  a  separate  armature  coil.  The  armature 
is  a  ring  of  soft  iron  wire,  widened  till  it  might  be  consid- 
ered a  short  hollow  cylinder.  Around  this  ring  are  coiled 
a  great  number  of  armature  coils,  as  shown  in  Fig.  152,  the 
ends  of  the  coils  being  brought  to  divisions  of  the  commu- 
tator. The  commutator  consists  of  a  number  of  plates 
radially  arranged  around  the  axis  of  the  armature,  and 
insulated  from  each  other.  The  commutator  divisions 
are  seen  on  the  right  of  the  armature  in  Fig.  153,  and  cor- 
respond in  number  to  the  armature  coils,  which  are  con- 
nected through  them  in  one  continuous  circuit. 

The  action  of  the  Gramme  may  be  easily  understood 
from  the  rules  of  Note  59.  In  Fig.  152  the  positive  direc- 
tion of  the  lines  offeree  is  from  A^to  S,  the  lines  entering 
the  ring  opposite  N,  and  dividing,  running  through  each 
half  of  the  ring  to  that  part  opposite  S,  where  they  leave 
the  ring  and  pass  to  S.  The  poles  N  and  6*  cannot  be 
considered  as  points,  and  the  lines,  therefore,  enter  the 
ring  all  along  its  lower  portion  (as  shown  in  the  figure) 
and  emerge  along  the  upper  part.  A  coil  in  the  position 
E"  has,  therefore,  the  maximum  number  passing  through 
its  plane.  If,  now,  the  armature  is  rotated,  so  that  E" 
passes  towards  E,  it  continually  incloses  a  decreasing 
number  of  lines  of  force,  and  a  direct  current  viewed  from 
N  is  induced.  The  E.  M.  F.  varying  as  the  rate  of  change, 


108  NOTES   ON 


is  zero  at  E"  and  a  maximum  at  a  point  opposite  S,  where 
the  coil  cuts  all  the  lines  at  right  angles.  As  the  rotation 
of  the  armature  continues,  the  coil  after  leaving  E  in- 
closes an  increasing  number  of  lines  of  force,  and  the 
current  is  therefore  inverse  as  viewed  from  N.  But 
from  E  to  E'  the  side  of  the  coil  viewed  is  the  oppo- 
site of  that  seen  from  E"  to  E,  and  the  inverse  current 
in  the  quadrant  from  E  to  E'  is  therefore  in  the  same 
absolute  direction  in  the  coil  as  the  direct  from  E"  to 
E.  Throughout  the  half  revolution  from  E"  to  £', 
therefore,  the  induced  current  flows  in  the  same  direc- 
tion, being  strongest  when  the  coil  is  nearest  the  pole 
S.  By  connecting  all  the  coils  in  series,  the  E.  M.  F. 
in  the  circuit  becomes  the  sum  of  all  in  the  individual 
coils,  and  as  these  occupy  all  possible  positions  at  any 
instant,  the  total  electromotive  force  is  constant,  the 
machine  thus  yielding  an  almost  absolutely  constant 
current. 

The  action  during  the  other  half  of  the  revolution 
may  be  traced  in  the  same  way.  The  coil  in  moving 
from  E'  to  N  incloses  a  decreasing  number,  inducing  a 
direct  current,  which  is  opposite  in  direction  to  that 
in  the  quadrant  from  E  to  E'.  If,  therefore,  in  the  latter 
the  current  had  flowed  away  from  the  point  E'  towards 
E,  it  would  in  E'N  flow  away  from  E'  towards  N,  and 
although  the  absolute  direction  of  the  currents  is  dif- 
ferent they  combine  to  lower  the  potential  of  E'.  During 
the  quadrant  between  N  and  E"t  the  coil  incloses  an  in- 
creasing number,  and  consequently  has  an  inverse  current 
induced,  but  this  inverse  current  is  in  the  same  absolute 
direction  as  the  direct  in  the  preceding  quadrant.  If, 
therefore,  throughout  the  upper  half  of  the  revolution  the 
current  flows  away  from  E',  it  will  in  the  lower  half  of 
the  revolution  flow  away  from  E'also.  Throughout  the 


ELECTRICITY  AND    MAGNETISM.  109 

whole  revolution  the  effect  is  to  raise  the  potential  of  E" 
and  lower  that  of  £',  and  if  brushes  touch  the  commu- 
tator at  these  points  they  will  possess  a  difference  of  po- 
tential which  may  be  utilized  in  the  production  of  a  cur- 
rent through  an  external  circuit.  The  Gramme  machine 
has  been  the  subject  of  much  investigation,  and  its  action 
has  been  variously  explained.  The  most  general  ex- 
planation in  any  case  of  electromagnetic  induction  is 
that  obtained  from  a  consideration  of  the  lines  of  force, 
and  this  is  the  one  adopted  by  Prof.  Thompson,  which 
has  only  been  given  here  in  slightly  greater  detail. 

The  armature  cylinder  is  made  of  soft  iron  wire,  both 
to  facilitate  the  rapid  magnetization  and  demagnetization, 
and  to  prevent  heating  from  the  Foucault  currents  which 
would  take  place  if  solid  metal  were  used.  The  change 
of  direction  of  the  lines  of  force  of  the  field  by  those  due 
to  the  current  is  frequently  very  marked  in  the  Gramme 
machine,  M.  Breguet  having  found  it  necessary  to  ad- 
vance the  commutator  brushes  70°  when  working  with  a 
Gramme  at  1770  revolutions.  As  the  internal  resistance 
of  the  Gramme  is  generally  small,  it  is  specially  adapted 
for  working  a  single  powerful  arc  light,  while  the  steadi- 
ness of  its  current  renders  it  well  adapted  for  incandescent 
lighting. 

73.  The  Brush  Machine  (§  411). 

This  machine  has  received  its  main  development  in  the 
United  States,  but  is  now  extensively  used  throughout  the 
world.  It  contains  many  peculiar  features,  and  is  distinct- 
ly a  separate  type,  although  frequently  alluded  to,  espe- 
cially by  French  authorities,  as  a  modification  of  the 
Gramme.  The  general  appearance  of  the  machine  is 
shown  in  Fig.  27. 

The  first  noticeable  peculiarity  is  in  the  disposition  of 


TIO 


NOTES   ON 


ELECTRICITY  AND   MAGNETISM.  m 

the  four  field  magnets,  which  are  placed  so  that  the  arma- 
ture coils  pass  between  similar  poles.  The  magnets  are 
oval  in  cross  section,  and  are  furnished  with  large  pole 
pieces,  approaching  very  closely  on  each  side  to  the  arma- 
ture. The  armature  is  a  soft  iron  disc,  with  deep  circular 
furrows  cut  in  its  sides  to  break  the  continuity  of  the  sur- 
face and  thus  prevent  the  heating  of  the  metal  by  the  in- 
duction of  Foucault  currents.  On  the  periphery  of  the 
armature  of  the  small  machine  there  are  eight  coils,  the 
two  coils  diametrically  opposite  being  in  one,  but  coiled  in 
opposite  directions  (See  Fig.  30),  so  as  to  act  in  unison  in 
the  induction  of  currents.  The  coils  project  from  the  ar- 
mature as  seen  in  Fig.  27,  the  reason  assigned  being,  that 
the  fanning  of  the  air  thus  caused  prevents  overheating. 

The  commutator  consists  of  four  rings  each  split  into 
four  segments.  A  cross  sec- 
tion of  one  of  the  rings  is  as  in 
Fig.  28,  the  two  ends  of  one 
pair  of  coils  being  connected 
to  the  segments  marked  I,  I, 
which  are  insulated  from  the 
segments  2,  2.  When  the 
brushes  of  the  commutator 
touch  the  latter  the  coils  are 
cut  out  of  circuit.  These  cut- 
ting out  segments  in  the  dif-  Fi£-  2g- 
ferent  rings  of  the  commutator  are  so  placed  that  at  every 
instant  one  coil  is  cut  out  ;  the  connections  being  made 
so  that  a  coil  is  not  in  circuit  in  that  part  of  the  rev- 
olution when  no  current  is  being  induced  in  it.  Each  of 
the  four  brushes  presses  on  the  commutator  rings  of  two 
coils  not  adjacent.  Numbering  the  coils  on  the  armature 
I,  2,  3  and  4 (Fig.  30)  in  order,  the  brushes  Z?  and  B*  are  in 
circuit  with  coils  I  and  3  and  £*  and  B*  with  coils  2  and  4. 


112 


NOTES  ON 


As  the  armature  revolves  each  coil  successively  passes 
through  all  parts  of  the  field.  When  a  coil  is  midway 
between  the  dissimilar  magnet  poles,  at  the  highest  point 
of  its  revolution,  the  number  of  lines  of  force  inclosed  is  a 
maximum,  but  changes  so  slowly  that  for  this  portion  of 
the  revolution  the  induced  current  is  but  small,  and  the 
coil  is,  therefore,  cut  out.  As  the  coil  approaches  the  large 
pole  pieces  and  passes  between  them  the  rate  changes 
rapidly.  If  a  piece  of  soft  iron  be  placed  between  two 
powerful  similar  magnet  poles,  the  lines  of  force  pass  into 
it  almost  parallel  on  each  side,  and  a  coil  moved  along  the 

bar  cutting  them  perpen- 

dicularly,  has  a  high  rate 

of  change  in  the  number  of 
lines  inclosed,  and  conse- 
_  quently  a  high  electro- 
motive  force  induced. 
Thus  in  Figure  29,  as 
nearly  all  the  lines  of  both 
poles  pass  through  the  soft 
iron  between  a  and  b,  the 
coil  A  in  moving  with- 
in that  region  experiences 

but   little  change    in   the 

<,          number  inclosed,   but 

as    it    approaches     either 

end  the  rate  of  change  is 

very  great.     The   electro- 

Vl«'**  motive   force   is    thus  in- 

duced somewhat  suddenly,  but  the  efficacy  of  the  pe- 
culiar arrangement  of  poles  for  the  induction  of  a  high 
electromotive  force  is  evident.  A  comparison  between 
Figs.  29  and  30  shows  this  to  be  nearly  the  condition  ex- 
isting in  the  Brush  machine.  The  latter  figure  is  a  plan 


ELECTRICITY  AND   MAGNETISM.  113 

of  the  machine.  As  each  pair  of  coils  is  connected  to  a 
separate  commutator  ring,  the  study  of  the  connections  is 
necessary  to  understand  the  complete  working.  In  the 
figure,  L  represents  a  lamp  in  the  external  circuit.  Bl,  B*, 
Bz  and  B*  are  the  commutator  brushes,  and  the  rings  are 


Fig.  30. 

numbered  I,  2,  3,  4,  as  illustrating  the  way  in  which  the 
ends  of  the  coils,  taken  in  regular  order  around  the  arma- 
ture, are  connected  at  the  commutator.  The  currents  in- 
duced in  the  several  coils  at  any  one  instant  will  have  the 
following  circuits,  coil  4  being  supposed  to  be  cut  out  : 
Coil  i—  i,  B\  L,  X,  13*,  2,  B\  M,  B\ 


Coil  2— 


2,  B\ 


,  Bl  <*>  B*,LX,  B\ 
Coil  3—3,  B\  L,  Xt  B\  2,  B\  M,  B\    ' 
These  paths  are  the  same  except  in  the  armature  coils 
and  the  resultant  current  will  have  the  path 

Bl  <^>  B\  Z,  X,  B\  2  B*,  M,  B\ 

An  instant  later  coil  4  will  be  in  circuit  and  I  cut  out. 
The  resultant  current  then  flows 


3,  B\  L,  X, 


\  M,  B\  3. 


114  NOTES   ON 


The  E.  M.  F.  in  circuit  is  evidently  that  due  to  two  coils 
in  series,  and  the  internal  resistance  of  the  machine  is 
diminished  by  the  fact  that  there  are  always  two  of  the 
four  armature  coils  in  parallel  arc. 

The  gross  efficiency  of  the  Brush  machine  is  lower  than 
that  of  some  others,  but  it  possesses  the  great  advantage 
of  yielding  so  high  an  electromotive  force  as  to  be  able  to 
burn  forty  arc  lights  in  series,  a  feat  which  no  other 
machine  can  accomplish.  As  there  are  only  two  coils  in 
series  at  one  time,  the  resultant  electromotive  force  is  far 
from  constant,  and  the  fluctuations  are  so  great  as  to 
utterly  unfit  the  machine  for  incandescent  lighting  or 
other  purposes  requiring  a  constant  current.  The  E.  M.  F. 
of  the  largest  Brush  machine  is  2000  volts  and  the  current 
about  10  amperes. 

74.  Edison  Machine. 

The  Edison  machine  ($  411)  is  a  shunt  dynamo.  Its 
chief  peculiarities  are  its  long  cylindrical  magnets  ending 
in  remarkably  heavy  pole  pieces  almost  encircling  the 
armature,  and  the  peculiar  construction  of  the  armature 
itself.  Theoretical  investigation  and  experiment  both 
point  to  long  cylindrical  magnets  as  most  efficient ;  and  in 
a  shunt  dynamo,  in  which  there  is  a  perpetual  endeavor 
for  a  permanent  adjustment  of  the  strength  of  the  field  to 
the  necessities  of  the  case,  it  is  advantageous  to  have  mag- 
nets of  considerable  mass,  as  the  change  of  field  brought 
about  by  a  variation  in  the  strength  of  the  field  current  is 
thus  made  more  gradual.  The  large  pole  pieces  tend  to 
make  the  field  more  uniform,  and  thus  act  to  secure  a 
constant  rate  or  a  uniform  change  of  electromotive  force 
throughout  the  rotation. 

Edison  calls  his  large  machine  a  "steam  dynamo,"  the 
engine  and  dynamo  being  on  the  same  bed-plate.  It  is 


ELECTRICITY    AND    MAGNETISM.  115 

specially  designed  for  use  at  a  central  station  to  supply 
power  or  work  incandescent  lights  throughout  a  district  of 
a  city.  As  established  in  New  York,  the  whole  weight  of 
dynamo  and  engine  is  nearly  thirty  tons,  sixteen  of  which 
are  in  the  magnets  and  pole  pieces.  The  core  of  the 
armature  is  made  up  of  sheet  iron  discs,  separated  from 
each  other  by  tissue  paper  and  bolted  together.  This 
prevents  heat  currents.  Instead  of  wire,  the  armature 
circuit  is  made  of  heavy  copper  bars,  each  bar  being  in- 
sulated from  the  next  and  from  the  iron  core  by  an  air 
space.  The  bars  are  connected  together  at  each  end  of 
the  armature  by  copper  discs,  there  being  half  as  many 
discs  at  each  end  as  there  are  bars.  Each  disc  has  lugs 
formed  on  it  on  opposite  edges,  to  which  two  bars  are 
connected,  and  the  whole  being  bolted  together,  the  bars 
and  discs  form  one  continuous  circuit  of  wonderfully  low 
resistance,  the  total  armature  resistance  of  a  machine  sent 
to  London  being  .0032  of  an  ohm.  This  very  low  resist- 
ance is  necessary  from  the  fact  that  the  machine  is  in- 
tended to  work  1300  incandescent  lights,  each  of  about 
137  ohms,  in  parallel  arc.  The  external  resistance  would, 
therefore,  be  only  .095  ohms.  As  the  number  of  lamps  in 
circuit  changes,  the  resistance  in  the  magnet  coils,  which 
are  in  a  shunt  of  the  main  circuit,  is  regulated  so  as  to 
keep  a  practically  constant  electromotive  force,  and  each 
lamp  then  burns  with  the  same  intensity  under  all  condi- 
tions. Edison's  large  machine  gives  an  E.  M.  F.  of  no 
volts  and  an  ordinary  current  of  1000  amperes. 

75.  Alternate  Current  Machines. 

Alternate  current  machines  have  been  used  in  Europe 

to  a  considerable  extent  for  incandescent  lamps  and  the 

Jablochkoff  and  other  candles.    Almost  any  machine  yields 

alternate  currents  if  used  without  a  commutator,  but  most 


1 1 6  NOTES  ON 


alternate  current  machines  have  a  large  number  of  armature 
coils  which  pass  between  the  poles  of  a  system  of  opjx>,<  •<! 
magnets,  so  arranged  that  the  positive  lines  of  force  in  the 
field  pass  alternately  through  the  coils  in  opposite  direc- 
tions, many  machines  inducing  fifteen  or  twenty  currents 
in  each  direction  in  every  revolution.  By  placing  a  num- 
ber of  coils  in  series,  so  disposed  that  the  induced  cur- 
rent in  all  is  in  the  same  direction  at  any  instant,  a  high 
electromotive  force  may  be  induced.  Many  machines 
have  the  connections  of  the  coils  adjustable,  so  that  they 
may  be  arranged  either  in  series  or  in  arc,  thus  per- 
mitting an  adaptation  to  the  requirements  of  the  external 
circuit.  An  objection  to  alternate  current  machines  is 
that  the  frequent  reversals  of  current  induce  extra  cur- 
rents of  so  high  electromotive  force  as  to  be  dangerous. 
The  more  coils  there  are  in  the  machine,  the  higher  the 
coefficient  of  self-induction,  and  the  greater  the  velocity 
the  greater  the  induced  electromotive  force,  so  that  the 
extra  current  may  be  much  stronger  than  the  normal 
current  of  the  machine.  This  disadvantage  exists  with 
many  continuous  current  dynamos,  particularly  the  Brush, 
the  electromotive  force  of  which  is,  as  already  stated,  not 
only  high  but  also  very  variable.  Every  change  of  the 
normal  current  induces  an  extra  current,  of  greater 
strength  as  the  revolution  of  the  machine  is  more  rapid. 
Alternate  current  machines  are  less  economical  than  those 
generating  a  continuous  current. 


ELECTRICITY    AND  MAGNETISM.  117 


VIII.  ELECTRIC     MOTORS. 
76.  General  Principles  (§  375). 

In  describing  the  action  of  Ritchie's  electric  motor,  Prof. 
Thompson  abandons  the  method  pursued  elsewhere  in 
his  book,  that  of  a  consideration  of  the  lines  of  force  in 
the  field,  and  adopts  another  less  general  explanation, 
that  of  the  mutual  action  of  magnet  poles.  If  the  core  of 
the  coils  CD  were  of  some  non-magnetic  substance  the 
description  given  would  not  apply,  although  the  motor 
would  still  work.  The  two  coils  Cand  D  in  Fig.  141  are 
practically  one,  and  this  will  in  any  magnetic  field  tend  to 
place  itself  so  as  to  bring  its  own  lines  of  force  in  the 
same  direction  as  those  of  the  field.  The  lines  of  the 
field  pass  from  Nio  S,  and  if  those  due  to  the  current  in 
the  coil  are  opposite  in  direction,  the  coil  will  tend  to 
rotate  into  a  position  of  equilibrium,  but  just  before  this  is 
attained  the  rotation  shifts  the  connections  of  the  coil  in 
the  mercury  cups,  the  current  changes,  its  lines  are  again 
opposite  to  those  of  the  field  and  the  coil  continues  its 
rotation  through  another  semicircle.  Owing  to  the  con- 
tinued shifting  of  the  direction  of  the  current,  the  coil  is 
perpetually  in  unstable  equilibrium,  and  the  rotation  is 
continuous  in  the  endeavor  to  attain  equilibrium.  As 
shown  in  Note  36,  the  work  done  by  a  coil  in  a  half  rev- 
olution is  2CHA,  and  if  the  coil  is  of  n  turns  this  becomes 
iCnlfA.  This  experiment  well  illustrates  many  of  the 
principles  of  electric  motors,  and  particularly  that  of  the 
commutator. 


NOTES   ON 


77.  Electric  Transmission  of  Power  to  a  Distance 
(§  376). 

As  illustrated  more  fully  in  Note  78,  mechanical  energy 
may,  by  the  use  of  a  dynamo  machine,  be  converted  into 
electrical  energy,  and  be  transferred  into  mechanical 
energy  again  by  another  dynamo  at  a  distance.  The 
value  of  this  fact  depends  on  circumstances.  There  are 
very  many  cases  in  which  power  thus  obtained,  due  pri- 
marily to  a  water-fall  at  a  distance,  would,  in  spite  of  the 
great  loss  in  transmission,  be  more  economical  than  the 
same  power  generated  from  steam  on  the  spot.  If  in- 
candescent lighting  becomes  an  established  system  in 
cities,  as  started  by  Edison  in  New  York,  the  same  wires 
which  work  the  lamps  at  night  may  transmit  power  by 
day,  and  small  motors  may  be  used  on  the  lamp  circuits. 
The  readiness  with  which  electric  energy  in  the  form  of  a 
current  may  be  subdivided,  offers  great  advantages  for 
its  distribution  in  small  amounts  over  the  system  of  dis- 
tribution of  steam  power  by  a  complicated  system  of 
shafting  and  belting.  Another  point  in  which  electric 
motors  may  be  used  is  in  electric  railroads.  A  stationary 
steam  engine  is  vastly  more  economical  than  a  locomo- 
tive, both  in  wear  and  tear  and  in  consumption  of  fuel 
and  water.  Generating  the  required  power  by  stationary 
engines,  this  economy  may  be  pushed  to  the  utmost  by 
the  adoption  of  large  engines  of  the  most  approved  type; 
and  as  an  electric  motor  may  be  made  of  great  power  but 
of  little  weight,  the  injury  to  rolling  stock  and  road  bed 
resulting  from  the  use  of  heavy  locomotives  would  be 
prevented.  Stationary  engines  working  dynamos  may  be 
placed  as  needed  and  the  current  generated  be  trans- 
mitted through  the  rails  to  the  motor.  Several  such  roads 
have  been  constructed  for  short  distances,  and  the  ques- 


ELECTRICITY   AND   MAGNETISM.  119 

tion  of  their   development  is  merely  one  of  commercial 
economy. 

78.  Theory  of  Electric  Motors  (§  377). 

The  theory  of  electric  motors  first  propounded  by  Jacobi 
has  of  late  received  mathematical  development  from 
others,  and  has  been  the  subject  of  much  experiment. 
The  underlying  principle  is  that  referred  to  in  Note  76, 
that  if  a  current  be  passed  through  a  coil,  free  to  move  in  a 
magnetic  field,  the  coil  will  move  into  a  certain  position, 
and  in  moving  is  capable  of  doing  work.  This  principle 
is  the  converse  of  that  underlying  the  action  of  the  dynamo 
machine,  and  it  is  therefore  easy  to  see  that  a  machine 
which  will  generate  currents  when  worked,  will,  on  the 
other  hand,  work  when  a  current  is  sent  through  it,  the 
rotation  as  a  motor  being  in  the  opposite  direction  to  that 
as  a  generator  of  current.  By  using  two  machines  in  the 
same  circuit,  the  current  generated  by  one  will  cause  the 
other  to  work.  The  commercial  value  of  the  fact  is  deter- 
mined by  the  cost  of  the  power  given  off  by  the  second 
machine. 

If  C  be  the  current  in  circuit  and  E  the  E.  M.  F.  of  the 
generator — not  the  difference  of  potential  at  the  terminals 
of  the  generator  (See  Note  70) — the  electrical  work  of  the 
generator  is  CE.  The  motor  in  rotating  backwards  gen- 
erates a  current  in  the  circuit  in  the  opposite  direction  to 
that  of  the  generator,  thus  diminishing  the  current  in 
circuit.  This  follows  directly  from  the  principle  of  con- 
servation of  energy.  If  £  is  the  back  electromotive  force 
of  the  motor,  the  electrical  work  of  the  motor  is  Ce.  The 
mechanical  work  given  off  by  the  motor  can  never  equal 
this,  as  the  motor  is  not  a  perfect  vehicle  for  the  trans- 
mission of  electrical  into  mechanical  energy,  but  in  calcu- 


120  NOTES  ON 


lation  the  motor  may  be  assumed  to  be  perfect,  and  correc- 
tions applied  to  the  final  results. 

The  ratio  of  the  work  of  the  motor  to  that  of  the  gener- 
ator is 

Ce  _    _£ 
CE~  E    ' 

or  the  return  is  the  ratio  of  the  electromotive  forces  of  the 
two  machines.  If  these  are  exactly  similar  and  working 
with  fields  of  equal  intensity,  a  condition  not  holding  in 
practice,  the  ratio  of  s  to  E  is  that  of  the  velocities  of  the 
two  machines. 

As  the  motor  generates  an  inverse  current,  the  current 
in  circuit  is 


7?  being  the  total  resistance. 

The  energy  of  the  generator  is  expended  partly  as  heat 
in  the  circuit,  and  partly  as  work  in  the  motor 

or  CE  =  C*R  +  work    .....    (3) 

Work  =  CE  -  C*R. 


Differentiating  for  a  maximum, 


C4) 


The  maximum  work  is,  therefore,  done  by  the  motor 
when  its  velocity  of  rotation  is  such  as  to  reduce  the  cur- 
rent in  circuit  to  one-half  that  due  under  Ohm's  Law  to 
the  electromotive  force  of  the  generator  and  the  resistance 
in  circuit.  This  is  the  case  when  s  =  h  E.  The  return 

is  then  by  (i),  -     =  *. 


ELECTRICITY   AND    MAGNETISM. 


121 


From  these  equations  the  conditions  of  economy  are 
deducible.  If  the  only  consideration  is  that  the  motor 
should  do  the  most  work,  equation  (4)  gives  the  condition 
that  its  back  electromotive  force  should  be  one-half  that 
of  the  generator.  If,  however,  the  governing  condition  is 
that  the  motor  should  work  as  economically  as  possible, 
(i)  indicates  that  the  electromotive  force  of  the  motor 
should  nearly  equal  that  of  the  generator.  The  return  is, 
then,  greater  than  one-half,  but  (2)  shows  that  the  current 
is  reduced,  that  CE,  the  work  of  the  generator,  is  also  di- 
minished and  that  consequently  a  greater  proportion  of 
a  smaller  amount  of  work  is  transmitted.  The  governing 
consideration  is  whether  the  motor  should  do  as  much  work 
as  possible,  regardless  of  cost,  or  work  with  the  greatest 
economy,  regardless  of  the  amount  of  work  done. 

These  conditions  are  of  the  greatest  importance,  but  are 
somewhat  difficult  to  reconcile.     Prof.  Thompson  has  de- 
vised   a  graphic    illustration    A  K  D 
which     presents     them    very 
clearly.  Draw  AB  to  represent 
E,  the  electromotive  force  of 
the  generator,  and  on  it  con- 
struct a  square,  ADCB.     On 
AS  lay  off  from  B,  BF\.o  repre- 
sent proportionally  e,  the  elec- 
tromotive force  of  the  motor, 
and  on  BF  construct  a  square 
BLGF.      Through     G    draw 
FH  parallel  to  £C  and  KL  parallel  to  AB. 

Then  the  work  of  the  generator  is  CE      • — ^ — ^-^ 

y? 


work  of  the  motor  is  Ce  = 


R 


122 


NOTES  ON 


But  e  (E  —  e)  is  the  area  of  GLCH,  and  E  (E  -  £)  is 
the  area  of  AFHD.  These  areas  are,  therefore,  marked 
as  "electric  energy  expended"  and  "useful  work,"  and 
the  ratio  of  the  two  areas  is  the  return.  From  the 
construction  the  point  G  will  always  fall  on  the  diagonal 
BD,  approaching  D  more  nearly  as  F  approaches 
A.  The  area  GLCH  corresponding  to  the  useful  work 
will,  therefore,  always  be  inscribed  within  the  triangle 
BCD,  and  will  have  its  maximum  value  when  it  is  a  square 
as  in  Fig.  31.  G  is  then  midway  between  .Z?and  D,  and 
the  area  of  GLCH  is  one-half  that  of  AFHD.  From 
similar  triangles,  e  is  also  one-half  of  E.  This  demonstrates 
the  case  of  maximum  work  ;  that  of  maximum  efficiency 
A  K  D  is  evident  from  Fig.  32. 

The  lettering  and  construc- 
l_l  tion  are  the  same,  but  the 
value  of  8  has  been  increased. 
The  ratio  of  GLCH  to 
AFHD  is  greater,  although 
each  area  is  less  than  in  the 
preceding  case.  A  greater 
ratio  of  a  less  amount  of  en- 
ergy is  thus  shown  to  be 
Fig.  y^  transmitted.  In  the  last 

figure  the  area  GLCH  is  by  geometry  equal  to  AKGF, 
and  the  square  KDHG,  the  difference  between  them, 
represents,  therefore,  the  factor  C*R,  or  the  loss  by  heat. 
The  smaller  this  loss  the  greater  the  return.  Any  desired 
return  may  be  calculated  by  making  GHCL  the  required 
fraction  of  ADHF.  If  this  is  to  be  90  per  cent.  KDHG 
must  be  -fa  of  ADHFt  and  this  may  be  secured  by  making 


ELECTRICITY  AND   MAGNETISM.  123 

DH  h  of  DC,  or  what  is  the  same  thing  BF  '-ft  of  AS. 
The  geometrical  construction,  therefore,  gives  the  same 
result  as  that  already  obtained,  that  the  return  is  the  ratio 

of  the  electromotive  forces,  or  that  —  •  should  be  -ft. 

r, 

The  work  done  by  the  generator  being  CE  the  question 
arises  whether  it  is  best  to  increase  the  energy  by  using 
a  stronger  current  or  a  higher  electromotive  force.  The 
loss  by  heat  is  C*R,  and  this  by  (2)  is  equal  to 


R 

If  now  E  and  e  are  both  increased  by  the  addition  of  the 
same  numerical  quantity,  the  difference  (E  —  «),  and  con- 
sequently the  loss  by  heat,  is  the  same.  Calling  the  new 
values  E'  and  «'  the  amount  of  work  done  is  easily  calcu- 
lated. The  work  done  by  the  generator  is  now 

E'(E'-e')         E'  (E  -  e} 


and  that  by  the  motor, 

g'(£"-O        e'  (E  - 


R  R 


....     (6) 


The  original  work  of  the  generator  was .     (7) 

and  of  the  motor         »""  (8) 

As  E'  and  «'  are  greater  respectively  than  E  and  e,  a 
comparison  of  (5)  with  (7)  and  (6)  with  (8)  shows  that 
more  work  has  been  done  by  the  generator  and  more 
by  the  motor,  while  the  loss  by  heat  was  the  same. 
There  is,  therefore,  clearly  an  economy  in  using  high 


124  NOTES   ON 


electromotive  forces  both  in  generator  and  motor.  The 
use  of  high  electromotive  force  is,  however,  more  dan- 
gerous and  necessitates  better  insulation. 

The  above  sets  forth  the  general  conditions  of  trans- 
mission of  energy  by  electricity.  A  few  deductions  are 
easily  made.  Solving  (3) 


E    ± 


2R 

Equating  (9)  and  (2), 

E  T    ^/~£r 


From  (10)  and  (i) 


_     _     _     _     (JO) 


Return  =        = 


—        .....     (") 


(n)  shows  that  the  return  is  not  independent  of  the 
amount  of  work  done  by  the  motor,  the  return  diminish- 
ing, other  things  equal,  as  the  work  done  increases.  If, 
however,  the  return  is  wished  to  be  the  same,  it  may 
be  secured  by  making  the  work  done  vary  inversely  as 
the  resistance  through  which  it  is  transmitted. 

Denoting  the  return,  or  —  ,  by  K,  the  following  equa- 
tions are  readily  deduced. 
Work  of  generator  =  E  ^E  ~  E  ^  =  (i  -  K}  .  ~    .     (12) 


Work  of  motor  =  KCE  =  A'(i  -  K}  .  —     .     .     .     (13) 

K 


ELECTRICITY   AND    MAGNETISM.  125 


Loss   in   heat  =  (E  R  g)*  =  (i  -  Kf  .  ^  .     .    .     .  (14) 

These  equations,  due  to  Marcel  Deprez,  show  that  work 
and  loss  by  heat  remain  constant  whatever  the  resistance 

—  E* 

in  circuit  may  be,  if  E  is  made  to  vary  as  y ' R,  that  -=? 

may  be  constant.  By  an  increase,  therefore,  of  the  elec- 
tromotive force  of  the  generator,  the  same  amount  of 
work  may  be  transmitted  to  a  greater  distance. 

79.  Modifications  of  Theory  in  Practice. 

The  preceding  demonstration  is  entirely  theoretical, 
and  although  the  general  conditions  hold,  they  are  much 
modified  in  practice,  the  modifications,  moreover,  being 
all  unfavorable  for  economical  results.  Among  the 
causes  of  error  are  the  following :  no  dynamo  machine  or 
motor  is  a  perfect  device  for  transmitting  energy.  The 
work  CE  done  by  the  first  dynamo  is  less  than  that  re- 
ceived by  it,  and  the  motor  is  unable  to  transfer  the  whole 
quantity  Cs  into  mechanical  energy.  If  /'"represents  the 
gross  efficiency  of  the  generator  and /that  of  the  motor, 

CE 

the  work  done  on  the  generator  is  — —    ,  and  that  done  by 

I1 

the  motor  is/.  Cs.     The  commercial  return  is,  therefore, 

Mechanical  'work  done  by  motor        fCs  £ 

Power  applied  to  generator         '    CE     *'*      '   E ' 

F, 

Assuming  for  both  generator  and  motor  the  gross  effi- 
ciency of  90  per  cent.,  the  maximum  commercial  return 
is  .81  of  the  power  applied,  the  work  done  being  then  very 
small,  or  .9  of  the  theoretical  return  in  the  preceding  equa- 
tions. Another  loss  is  that  caused  by  leakage.  If  the 


126  NOTES  ON 


insulation  is  not  perfect,  leakage  occurs  all  along  the  line, 
and  the  current  at  the  motor  is  less  than  the  current  at 
the  generator.  This  loss  is  greater  as  the  electromotive 
force  is  raised.  If  the  motor  is  a  self-exciting  dynamo, 
the  weaker  current  causes  a  weaker  field  than  that  of  the 
generator,  so  that  if  the  two  machines  are  exactly  similar, 
the  ratio  of  the  electromotive  forces  are  not  the  same  as 
the  ratio  of  the  velocities.  There  may  be  other  causes 
operating,  but  these  are  sufficient  to  show  that  in  no  case 
can  the  theoretical  return  be  fully  realized.  Marcel  De- 
prez  has  at  different  times  made  experiments  in  transmit- 
ting power  to  a  distance.  The  most  careful  up  to  this 
date  were  made  in  Paris  on  March  4,  1883,  when  he  suc- 
ceeded in  obtaining  from  a  motor  4.439  H.  P.  (French), 
which  had  been  transmitted  through  a  resistance  of  160 
ohms,  corresponding  to  about  ten  miles  of  ordinary  tele- 
graph wire.  The  power  applied  to  the  generator  was 
12.267  H.  P.,  but  the  electrical  work  CE  of  the  gener- 
ator was  only  9.751  H.  P.  Gramme  machines  of  high 
resistance  were  used,  the  electromotive  force  of  the 
generator  being  2,480  volts  and  that  of  the  motor  1,779. 
The  electrical  return  was,  therefore,  71.7  per  cent.,  the 
commercial  return  36.2  per  cent. 

That  the  electric  transmission  of  energy  may  be  a  suc- 
cess, a  generator  is  necessary  which  with  a  constant 
speed  gives  a  constant  electromotive  force,  whatever 
changes  may  take  place  in  the  external  circuit.  If  a  gen- 
erator is  to  work  one  hundred  motors,  it  must  be  able  to 
work  them  all  at  once  or  a  few  at  a  time,  without  running 
risk  of  injury  from  sudden  changes  of  the  number  in  use. 
Machines,  or  combinations  of  machines,  for  this  purpose 
have  been  invented,  one  type  of  which,  compound  dynamos 
(Note  68,  i),  has  already  been  referred  to.  It  is  also  nec- 
essary that  the  motors  should  move  at  the  same  speed 


ELECTRICITY   AND    MAGNETISM.  127 

whether  doing  work  or  not.  Improvements  in  these  two 
points  will  make  the  transmission  of  power  a  commercial 
success. 

Go.  Peltier  Effect  (§  380). 

This  effect  is  caused  when  a  current  flows  from  one 
metal  to  another,  and  is  independent  of  the  resistance.  As 
stated  on  page  343,  the  heating  varies  directly  as  the  cur- 
rent, and  the  junction  which  is  heated  by  a  current  in  one 
direction  is  cooled  if  the  direction  of  the  current  is  re- 
versed. Letting  P  be  the  heat  in  joules  produced  at  a 
junction  of  two  metals  per  second  by  a  current  of  one 
ampere  : 

Total  heat  in  joules  =  C*R  ±  PC 
=  C(CR  ±  P). 

In  the  last  equation  the  quantity  in  brackets  and  the 
term  CR  are  both  electromotive  forces.  P  must,  there- 
fore, be  also  an  electromotive  force  measured  in  volts, 
although  commonly  called  the  "  coefficient  of  the  Peltier 
effect." 

By  carefully  measuring  the  change  of  temperature  at  a 
junction  with  the  current  alternately  in  opposite  directions  : 

//,  =  CV?  +  PC 
H*  =  C*R  -  PC 

Hi  -  H,  =  2PC        orP  =  Hl  ~H<i  . 

2  C 

If  the  current  is  one  ampere, 


Hi  and  H^  are  measured  in  joules,  or  in  ergs  x  io7, 
hence  the  numerical  value  of  P  may  readily  be  found  in 
ergs. 


128  NOTES  ON 


The  equation  in  the  Elementary  Lessons  was  written 
before  the  joule  had  come  into  use  as  a  practical  unit  of 
heat,  but  is  the  same  as  the  above,  since  the  joule  is  equal 
to  .24  of  a  water-gramme-degree  centigrade  thermal 
unit. 

81.  Secondary  Batteries  (§  415). 

It  has  been  found  that  lead  dioxide  is  highly  electro- 
negative to  metallic  lead,  the  difference  of  potential  between 
the  two  in  dilute  sulphuric  acid  being  about  2.7  volts. 
These  two  substances  are  used  in  the  secondary  battery, 
partly  on  account  of  the  high  difference  of  potential  they 
possess,  but  mainly  on  account  of  the  facility  with  which 
the  lead  dioxide  may  be  formed  by  electrolysis.  The  Plante 
cell  consists  of  two  plates  of  lead  in  dilute  sulphuric  acid. 
If  a  current  is  passed  through  the  cell,  the  liquid  is  decom- 
posed, hydrogen  is  evolved  on  the  kathode  and  oxygen 
on  the  anode.  The  latter  unites  chemically  with  the  lead, 
forming  lead  dioxide,  and  this  being,  as  stated,  highly 
electro  negative  to  lead,  if  the  original  source  of  electricity 
be  removed  and  the  secondary  cell  short  circuited  a  cur- 
rent will  flow  through  the  cell  in  the  opposite  direction  to 
that  of  the  charging  current,  and  will  in  time  deoxidize 
the  negative  plate.  The  cell  is  then  discharged.  The 
process  of  charging  is,  therefore,  merely  one  of  polariza- 
tion, and  the  effect  of  the  current  which  it  is  of  the  most 
importance  to  avoid  in  the  ordinary  cell  is  the  basis  of  the 
utility  of  the  secondary  battery.  Charged  in  this  way, 
however,  a  Plante"  cell  yields  but  little  current.  In  prac- 
tice the  cell  is  charged  as  above,  discharged  and  then 
charged  in  the  opposite  direction,  and  this  alternate 
charging  in  opposite  directions  in  time  renders  both  plates 
spongy  or  cellular  in  texture,  enabling  the  oxygen  given 
up  at  the  anode  to  more  readily  enter  into  combina- 


ELECTRICITY   AND    MAGNETISM.  1 29 

tion  with  the  lead  and  forming  a  dioxide  layer  of  greater 
thickness. 

The  Faure  and  later  secondary  batteries  are  similar  in 
principle  to  the  Plants,  but  have  the  plates  at  the  begin- 
ning coated  with  lead  oxide.  When,  then,  a  current  is 
passed  the  anode  is  oxidized  to  lead  dioxide,  and  the 
kathode  deoxidized  to  metallic  lead  by  the  hydrogen 
evolved.  The  cell  does  not  need  the  preliminary  treat- 
ment of  the  Plante",  but  is  ready  for  use  immediately  after 
the  first  charging,  but  the  chemical  action  is  much  more 
complex,  and  the  cell  is  probably  not  so  durable. 

The  term  "  storage  of  electricity  "  is  frequently  used  in 
connection  with  secondary  batteries,  but  is  not  strictly 
accurate,  as  the  portion  of  the  energy  of  the  charging  cur- 
rent which  is  stored  in  the  cell  is  in  the  form  of  energy  of 
chemical  separation,  and  is  again  transformed  into  elec- 
trical energy  when  the  circuit  is  closed.  The  utility  of  the 
secondary  battery  arises  from  the  fact  that  the  chemical 
action  when  the  circuit  is  open,  is  not  great,  and  that  the 
cell  may  be  used  after  an  interval  of  a  few  days  from  the 
time  it  was  charged.  The  objection  to  its  use  is  that,  in 
in  the  first  place,  only  a  portion  of  the  electrical  energy  of 
the  current  can  be  stored  in  the  cell  as  chemical  energy, 
and  secondly,  that  chemical  action  does  take  place  to  a 
certain  extent  when  the  cell  is  not  in  use,  and  that  it  can- 
not, therefore,  store  energy  indefinitely.  The  chief  dete- 
rioration arises  from  the  formation  of  lead  sulphate.  A 
cell,  moreover,  wears  out  eventually  and  becomes  prac- 
tically useless.  There  are  many  ways  in  which  secondary 
batteries  may  be  of  service,  particularly  in  connection 
with  dynamos  in  electric  lighting,  but  the  anticipations  of 
their  sphere  of  usefulness  entertained  shortly  after  the  in- 
troduction of  Faure 's  battery  were  exaggerated. 


13°  NOTES   ON 


IX.  TELEGRAPHY  AND  TELEPHONY. 
82.  The  Morse  Alphabet  (§  425). 

The  alphabet  printed  on  p.  397  is  the  international 
alphabet  used  in  Europe  and,  in  fact,  everywhere  except 
in  the  United  States  and  Canada,  where  the  code  origi- 
nally introduced  by  Morse  is  still  in  use.  The  international 
is  probably  the  better,  as  it  is  more  easy  to  distinguish 
combinations  of  letters  and  avoid  mistakes,  but  it  is  ex- 
tremely difficult  to  make  any  change  in  a  code,  however 
faulty  it  may  be,  when  it  has  once  come  into  use.  The 
alphabet  used  in  America  is  as  follows  : 

A T  

B U 

C---  V 

D W 

E  -  X 

F Y  -  -    -  - 

Q 2 

H &  - 

T         ___     _  2         —     —     —     — 

L  4 

M 5 

N 6 

O  -  7 

p 8 

Q 9 

R  _    _  _  o  

S 


ELECTRICITY  AND   MAGNETISM.  131 

83.  American  System  of  Telegraphy*  (§  426). 

The  European  system  is  known  as  the  "open  circuit" 
system,  the  current  flowing  only  when  the  key  is  depressed. 
Many  inconstant  cells  like  the  Leclanche"  may,  therefore, 
be  used.  In  America,  however,  the  current  flows  continu- 
ously when  no  messagejs  passing.  When  an  operator 
wishes  to  telegraph  he  first  breaks  the  circuit  by  a  switch 
attached  to  the  key,  and  then  makes  the  signals,  the  cir- 
cuit being  closed  when  the  key  is  depressed.  When  not 


Fig.  33— KEY. 

telegraphing  the  switch  must  be  closed  or  no  signals  can 
be  made  by  any  other  operator  on  the  line.  The  general 
appearance  of  the  American  key  is  shown  in  Fig.  33.  The 
key  is  fastened  to  the  table  by  the  screws  B  and  L,  the 
former  being  insulated  from  the  metal  of  the  key,  the 
latter  in  connection  with  it.  One  wire  is  connected  to 
the  metal  of  the  key,  generally  at  L,  and  the  other  clamped 
by  B.  The  switch  moves  horizontally,  and  when  pushed 
towards  the  left  in  the  figure,  makes  contact  with  B 

*  Only  a  mere  outline  of  the  closed  circuit  system  is  here  given.  Full 
information  may  be  found  in  books  on  telegraphy,  the  best  being  prob- 
ably Prescott's  u  Electricity  and  the  Electric  Telegraph." 


132 


NOTES   ON 


and  connects  it  with  L.  When  pushed  to  the  right  the 
circuit  is  open,  and  is  closed  only  when  the  key  is  depressed 
and  contact  made  with  the  head  of  the  screw  B. 

The  general  arrangement  of  apparatus  at  a  way  station 
is  shown  in  Fig.  34.  The  current  entering  by  the  line 
wire  on  the  right  first  passes  through  the  key  A",  the  switch 


Fig.  34- 


in  the  position  shown  touching  the  head  //"of  the  screw  B 
(Fig.  33),  and  closing  the  circuit.  From  the  key  it  passes 
to  the  relay  R,  entering  at  the  binding  post  A,  passing 
around  the  electromagnet^/,  and  issuing  at  B,  passing  into 
the  line  to  the  next  station.  This  current  is  furnished 
either  by  a  powerful  battery  at  one  end  of  the  line,  or  by  a 
battery  at  each  end,  acting  in  the  same  direction.  In 
front  of  the  electromagnet  M is  a  vibrating  lever  of  iron  or 
one  furnished  with  an  iron  armature,  pivoting  at  the  point 
P.  When  the  current  passes,  this  lever  touches  the  stop 
D  and  closes  the  local  circuit  DXLSYP  through  the 


ELECTRICITY   AND    MAGNETISM. 


133 


"sounder"  S.  When  the  line  current  ceases,  the  lever  V 
is  drawn  back  by  the  spring  and  contact  at  D  is  broken. 
Whenever,  therefore,  an  operator  at  any  station  opens  his 
switch  and  signals,  every  relay  on  the  line  works,  and 
each  relay  works  a  "sounder"  through  the  intervention  of 
its  local  battery.  As  the  current  always  runs  in  the  same 


35-—  RELAY. 


direction,  the  relay  works  for  every  signal  from  which  evei 
way  it  may  come.  In  the  open-circuit  system  a  relay  is 
necessary  for  messages  in  each  direction.  The  Western 
Union  relay  is  shown  in  Fig.  35, 

The  printing  receiver  or  embosser  is  but  little  used  in 
America,  messages  being  read  by  sound.    The  "sounder" 


Fig.  36.— SOUNDER, 
consists  of  two  electromagnets  which  attract  an  armature 


134  NOTES   ON 


whenever  the  local  current  passes.  The  armature  is  at- 
tached to  a  lever,  which  makes  a  sharp  click  by  striking 
against  a  stop  whenever  the  armature  moves.  After  a 
little  practice  the  operator  can  read  the  message  easily. 

84.  Faults  (§  427). 

Formulas  may  be  easily  worked  out  for  determining 
the  position  of  a  fault,  on  the  supposition  that  the  resistance 
of  the  fault  is  itself  constant.  In  practice,  this  is  seldom 
the  case,  and  never  so  in  submarine  cables,  as  the  cur- 
rent escaping  at  the  fault  causes  electrolysis  of  the  sea 
water,  either  depositing  chloride  of  copper  over  the  fault, 
or  clearing  away  such  deposit  according  as  the  current  is 
positive  or  negative.  The  exact  determination  of  the  po- 
sition of  faults  requires,  therefore,  great  skill  in  making 
the  tests  and  good  judgment  in  interpreting  the  results 
obtained. 

85.  Simultaneouss  Transmission  (§  428). 

This  method  generally  requires  the  use  of  a  polarized 
relay.  That  of  Siemens  is  probably  the  most  easily  un- 
derstood. The  simple  form  of  the  relay  is  shown  in  Fig. 
37.  5  is  the  south  pole  of  a  steel  magnet  bent  at  a  right 
angle.  The  lever  aD  is  of  soft  iron  pivoted  at  D  and  is 
of  south  polarity.  Attached  to  the  north  pole  of  the  steel 
magnet  are  two  soft  iron  cores  n  and  n',  around  which  is 
coiled  wire  in  the  same  circuit  but  in  opposite  directions. 
When  no  current  passes  the  cores  are  of  north  polarity, 
and  the  oscillating  lever  aD  is  attracted  to  the  one  near- 
est it.  If  a  current  is  sent  through  the  coils,  the  cores  be- 
come electromagnets  of  opposite  polarity,  and  the  lever 
then  moves  towards  the  north  pole.  If  the  current  is  re- 
versed the  lever  moves  in  the  opposite  direction,  and  if 
the  circuit  is  broken  the  lever  moves  towards  the  nearest 


ELECTRICITY   AND    MAGNETISM. 


135 


core,  as  both  then  become  north  poles  from  the  inductive 
action  of  the  steel  magnet,    The  motion  of  the  lever  is  con- 


9  TO  LINE 


TO  LOCAL 
BATTERY 


DTO  LINE 


Fig.  37' 

trolled  by  the  two  studs  at  a,  the  upper  of  which  is  con- 
nected with  a  local  battery  in  circuit  with  a  sounder,  as  in 
the  common  relay.  The  position  of  these  studs  is  so  regu- 
lated that  the  lever,  even  when  touching  the  upper  one  and 
closing  the  local  circuit,  is  nearer  ri  than  n.  The  moment, 
therefore,  that  the  line  current  ceases,  the  lever  is  attracted 
by  two  north  poles,  but  moves  towards  the  nearest,  break- 
ing the  local  circuit  at  the  stud.  No  springs  are  necessary, 
nor  does  the  relay  require  any  adjustment  for  strength  of 
current.  From  the  foregoing  it  is  seen  that  when  the 
direction  of  the  current  is  such  as  to  make  «  a  north  pole 
and  n>  a  south,  the  lever  moves  and  closes  the  local  circuit. 
When  no  current  passes  in  the  line,  or  when  it  passes  in 
the  opposite  direction,  the  local  circuit  is  open. 

In  sending  two  messages  at  the  same  time  in  the  same 
direction,  two  keys  are  used,  one  reversing  the  current,  send- 
ing positive  or  negative  currents,  the  other  sending  weak 
or  powerful.  The  strength  of  the  current  is,  therefore, 
controlled  by  one  key,  its  sign  by  the  other.  The  method 


i36 


NOTES   ON 


used  by  Edison  for  transmitting  is  shown  in  the  figure.    In 
the  position  shown  the  battery  B  has  its  terminals  at  N 
L1NE  and   Pt  the   current  passing 

from  B  through  A"2  to  the 
spring  6*  and  thence  to  P,  If 
the  key  K'  is  worked  cur- 
rents of  either  polarity  may 
be  sent  into  the  line,  and 
passing  through  a  polarized 
relay  at  the  receiving  station, 
a  sounder  in  the  local  circuit 
is  \vorked  whenever  a  cur- 
rent in  a  given  direction  is 
transmitted  by  A".  The 
strength  of  the  current  is 
immaterial,  the  polarized 


Fig.  33. 


relay  answering  only  to  cur- 


rents in  one  direction.  As  shown  in  the  figure,  the  circuit 
of  the  battery  B\  which  is  much  larger  than  B,  is  open. 
If,  however,  the  key  /v"2  is  depressed  the  spring  6"  comes  in 
contact  with  the  point  m  and  breaks  contact  with  #,  and 
as  it  is  separated  from  A"2  by  the  insulating  material  /,  the 
current  of  B  now  has  to  pass  through  B'  m  and  5  to  P, 
and  is,  of  course,  reinforced  by  the  the  powerful  current 
of  B'  in  the  same  direction.  Whenever  A"3  is  depressed, 
therefore,  the  points  N  and  P  retain  their  polarity,  but  the 
current  is  of  three  or  four  times  its  original  strength.  In 
practice  all  contacts  are  made  by  springs,  so  that  the  cir- 
cuit is  never  broken  at  A%,  but  one  current  is  followed  di- 
rectly by  the  other.  The  message  transmitted  by  K '8  is 
received  by  an  ordinary  relay  in  the  same  circuit  with  the 
polarized  relay  at  the  receiving  station,  the  lever  of  which 
is  controlled  by  a  spring  so  adjusted  that  the  weak  cur- 
rent of  B  will  not  cause  sufficient  magnetism  in  the  elec« 


ELECTRICITY  AND   MAGNETISM.  137 

tromagnets  to  attract  the  armature  against  the  action  of 
the  spring,  but  when  A%  is  worked  the  current  due  to 
B  +  B'  easily  overcomes  it,  whether  the  current  be  posi- 
tive or  negative,  and  the  relay,  therefore,  transmits  all 
signals  made  by  K *. 

The  quadruplex  is  merely  an  extension  of  the  duplex, 
using  the  diplex  or  double  transmission.  If  in  Fig.  163 
(Thompson)  the  transmitting  apparatus  just  described  is 
used  instead  of  the  keys  R  and  R,  and  if  between  A  and 
B  two  relays  are  placed  in  series,  one  an  ordinary  relay 
and  the  other  a  polarized,  the  figure  would  represent  the 
general  arrangement  of  Edison's  quadruplex  system  widely 
used  in  the  United  States. 

86.  Blake's  Transmitter. 

In  most  telephone  circuits,  the  receiving  instrument  is  a 
Bell  telephone,  but  the  transmitting  is  a  modification  of 
Edison's  telephone,  known  as  Blake's  Transmitter.  The 
waves  of  sound  impinge  on  a  metallic  diaphragm,  caus- 
ing it  to  press  with  more  or  less  force  on  a  carbon  button. 
A  current  from  a  battery  passes  through  the  button  and 
the  varying  pressure  of  the  diaphragm  causes  a  varying 
resistance  in  the  circuit,  and  produces  in  the  current  fluc- 
tuations, corresponding  in  number  and  time  to  the  waves  of 
sound.  If  this  current  is  passed  through  a  Bell  telephone, 
the  message  could  be  heard.  As  now  used,  however,  the 
battery  circuit  is  entirely  local.  In  this  local  circua  is  the 
primary  coil  of  an  induction  coil,  the  secondary  being  in 
circuit  with  the  line  to  the  next  station.  Every  fluctua- 
tion, therefore,  in  the  strength  of  the  local  circuit,  due  to 
the  change  of  pressure  on  the  carbon  button  of  the  trans- 
mitter, induces  a  current  in  the  secondary  coil  which 
works  a  Bell  telephone  at  the  distant  station.  The  induc- 
tion coil  is  small,  but  it  causes  the  electromotive  force  of 


I38  NOTES  ON 


the  line  circuit  to  be  much  greater  than  that  due  to  the 
battery  and  extends  the  use  of  the  telephone  to  greater 
distances. 

87.  Telephone  Exchanges. 

The  use  of  the  telephone  has  been  greatly  extended  by 
the  system  of  exchanges.  A  large  number  of  persons 
have  telephoae  circuits  to  a  central  office,  where  any  two 
circuits  may  be  joined,  thus  enabling  any  two  to  converse. 
A  great  difficulty  in  all  telephone  circuits  is  due  to  induc- 
tion. The  instrument  is  so  extremely  delicate  that  any  in- 
constant current  near  it  induces  sufficiently  powerful  cur- 
rents in  the  telephone  circuit  to  frequently  obliterate  a 
message  entirely.  Telegrams  may  be  read  in  telephones 
if  the  telegraph  and  telephone  circuits  approach  each 
other  very  closely,  and  telephone  messages  may  also  be 
heard  in  other  circuits  than  that  in  which  they  are  trans- 
mitted. Most  of  the  disturbances  commonly  attributed  to 
induction  are,  however,  in  all  probability  due  to  grounded 
telegraph  circuits. 


ELECTRICITY    AND    MAGNETISM 


139 


REFERENCES  TO  PROF.  THOMPSON'S  ELEMENTARY  LESSONS. 


191,  Note  26. 

§353, 

Note  45,  46,  47. 

ig2, 

34- 

§357, 

"     48. 

199, 

I. 

§358, 

49,  50- 

200, 

i. 

§36o, 

Si- 

201, 

2. 

§  36i, 

"     52. 

202, 

3- 

§  362, 

53- 

203, 

4- 

i364' 

54- 

204, 

5- 

§  367, 

55- 

237, 

6,  7,  8,  g,  10. 

§  371, 

44     56. 

238, 

ii. 

§  372, 

57- 

239, 

12. 

§  374, 

44     58- 

24O, 

13- 

§  375, 

"     76. 

241, 

14. 

§376, 

77- 

245, 

§  377, 

78,  79- 

246, 

ii' 

§378, 

"     55- 

247, 

17. 

§  380, 

"     80. 

252, 

18. 

§  39i, 

258, 

ig. 

§  392, 

"     59- 

26l, 

20,  21. 

§  393, 

262, 

51,  c. 

§  394, 

'  "     59,  60. 

310, 

22,  23,  24,25,  26, 

§  395, 

"     61. 

27. 

§  396, 

41      62. 

311, 

28. 

§  397, 

"      63. 

312, 

23. 

§398, 

"     66. 

313, 

2g. 

§404, 

"     64. 

3M, 

30. 

8  405, 

65. 

315, 
3i6, 

3i,  32. 

33- 

§407, 

i  "     67,  68,  69, 

317, 

33- 

|  409, 

71. 

3i8, 

34,  35,  36,  37,  38. 

i  4IO> 

41      72. 

319, 

36. 

§4ii, 

73,  74,  75. 

320, 

39- 

§  415, 

44      81. 

324, 

40. 

§  425, 

44      82. 

325^, 

41. 

ts  426, 

"     83. 

327, 

42. 

§427, 

'4     84. 

338, 

42. 

§428, 

44      85. 

43- 

§  436, 

44      86,  87. 

352,' 

44- 

14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 


In*     \y    fl  1    ^1                  Ib^  Lw    1                If 

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f    iditt'a/Rfisr 

' 

JAN  2    1957 



^.lillSSRK66               u-uSaSttoSKmi. 

Berkeley 

Murdock,  J. 

B. 

M8 

Notes  on 

sleotricity 

ana  magnet  1 

sm 

THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


